tag:blogger.com,1999:blog-73996219456088611432024-03-13T23:15:18.637-07:00Discrete Mathematics Notes - DMSDiscrete maths notes for academicsSumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-7399621945608861143.post-345236135591299492009-01-09T23:02:00.000-08:002009-01-09T23:06:34.504-08:00LOGIC AND SETS<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///C:%5CUsers%5Csujata%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///C:%5CUsers%5Csujata%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///C:%5CUsers%5Csujata%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:trackmoves/> <w:trackformatting/> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:donotpromoteqf/> <w:lidthemeother>EN-IN</w:LidThemeOther> <w:lidthemeasian>X-NONE</w:LidThemeAsian> <w:lidthemecomplexscript>X-NONE</w:LidThemeComplexScript> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> <w:splitpgbreakandparamark/> <w:dontvertaligncellwithsp/> <w:dontbreakconstrainedforcedtables/> <w:dontvertalignintxbx/> <w:word11kerningpairs/> <w:cachedcolbalance/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathpr> <m:mathfont val="Cambria Math"> <m:brkbin val="before"> <m:brkbinsub val="--"> <m:smallfrac val="off"> <m:dispdef/> <m:lmargin val="0"> <m:rmargin val="0"> <m:defjc val="centerGroup"> <m:wrapindent val="1440"> <m:intlim val="subSup"> <m:narylim val="undOvr"> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" defunhidewhenused="true" defsemihidden="true" defqformat="false" defpriority="99" latentstylecount="267"> <w:lsdexception locked="false" priority="0" semihidden="false" unhidewhenused="false" qformat="true" name="Normal"> <w:lsdexception locked="false" priority="9" semihidden="false" unhidewhenused="false" qformat="true" name="heading 1"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 2"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 3"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 4"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 5"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 6"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 7"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 8"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 9"> <w:lsdexception locked="false" priority="39" name="toc 1"> <w:lsdexception locked="false" priority="39" name="toc 2"> <w:lsdexception locked="false" priority="39" name="toc 3"> <w:lsdexception locked="false" priority="39" name="toc 4"> <w:lsdexception locked="false" priority="39" name="toc 5"> <w:lsdexception locked="false" priority="39" name="toc 6"> <w:lsdexception locked="false" priority="39" name="toc 7"> <w:lsdexception locked="false" priority="39" name="toc 8"> <w:lsdexception locked="false" priority="39" name="toc 9"> <w:lsdexception locked="false" priority="35" qformat="true" name="caption"> <w:lsdexception locked="false" priority="10" semihidden="false" unhidewhenused="false" qformat="true" name="Title"> <w:lsdexception locked="false" priority="1" name="Default Paragraph Font"> <w:lsdexception locked="false" priority="11" semihidden="false" unhidewhenused="false" qformat="true" name="Subtitle"> <w:lsdexception locked="false" priority="22" semihidden="false" unhidewhenused="false" qformat="true" name="Strong"> <w:lsdexception locked="false" priority="20" semihidden="false" unhidewhenused="false" qformat="true" name="Emphasis"> <w:lsdexception locked="false" priority="59" semihidden="false" unhidewhenused="false" name="Table Grid"> <w:lsdexception locked="false" unhidewhenused="false" name="Placeholder Text"> <w:lsdexception locked="false" priority="1" semihidden="false" unhidewhenused="false" qformat="true" name="No Spacing"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 1"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 1"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 1"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 1"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 1"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 1"> <w:lsdexception locked="false" unhidewhenused="false" name="Revision"> <w:lsdexception locked="false" priority="34" semihidden="false" unhidewhenused="false" qformat="true" name="List Paragraph"> <w:lsdexception locked="false" priority="29" semihidden="false" unhidewhenused="false" qformat="true" name="Quote"> <w:lsdexception locked="false" priority="30" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Quote"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 1"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 1"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 1"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 1"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 1"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 1"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 1"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 1"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 2"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 2"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 2"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 2"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 2"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 2"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 2"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 2"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 2"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 2"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 2"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 2"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 2"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 2"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 3"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 3"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 3"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 3"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 3"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 3"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 3"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 3"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 3"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 3"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 3"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 3"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 3"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 3"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 4"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 4"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 4"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 4"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 4"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 4"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 4"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 4"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 4"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 4"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 4"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 4"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 4"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 4"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 5"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 5"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 5"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 5"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 5"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 5"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 5"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 5"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 5"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 5"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 5"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 5"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 5"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 5"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 6"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 6"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 6"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 6"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 6"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 6"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 6"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 6"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 6"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 6"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 6"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 6"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 6"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 6"> <w:lsdexception locked="false" priority="19" semihidden="false" unhidewhenused="false" qformat="true" name="Subtle Emphasis"> <w:lsdexception locked="false" priority="21" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Emphasis"> <w:lsdexception locked="false" priority="31" semihidden="false" unhidewhenused="false" qformat="true" name="Subtle Reference"> <w:lsdexception locked="false" priority="32" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Reference"> <w:lsdexception locked="false" priority="33" semihidden="false" unhidewhenused="false" qformat="true" name="Book Title"> <w:lsdexception locked="false" priority="37" name="Bibliography"> <w:lsdexception locked="false" priority="39" qformat="true" name="TOC Heading"> </w:LatentStyles> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-1610611985 1107304683 0 0 159 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} @font-face {font-family:CMBX10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMMI10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMCSC10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMSY10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR7; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMTI10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:MSBM10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} a:link, span.MsoHyperlink {mso-style-noshow:yes; mso-style-priority:99; color:blue; mso-themecolor:hyperlink; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-noshow:yes; mso-style-priority:99; color:purple; mso-themecolor:followedhyperlink; text-decoration:underline; text-underline:single;} span.EmailStyle17 {mso-style-type:personal; mso-style-noshow:yes; mso-style-unhide:no; mso-ansi-font-size:11.0pt; mso-bidi-font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; color:windowtext;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} @page Section1 {size:612.0pt 792.0pt; margin:72.0pt 72.0pt 72.0pt 72.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;} </style> <![endif]--> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:14;" >
<br /><o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >1.1. Sentences<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >In this section, we look at sentences, their truth or falsity, and ways of combining or connecting sentences<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >to produce new sentences.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >A sentence (or proposition) is an expression which is either true or false. The sentence \2 + 2 = 4" is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >true, while the sentence \</span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational" is false. It is, however, not the task of logic to decide whether<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >any particular sentence is true or false. In fact, there are many sentences whose truth or falsity nobody<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >has yet managed to establish; for example, the famous Goldbach conjecture that \every even number<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >greater than 2 is a sum of two primes".<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >There is a defect in our de_nition. It is sometimes very di_cult, under our de_nition, to determine<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >whether or not a given expression is a sentence. Consider, for example, the expression \I am telling a<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >lie"; am I?<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Since there are expressions which are sentences under our de_nition, we proceed to discuss ways of<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >connecting sentences to form new sentences.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Let </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >denote sentences.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >(CONJUNCTION) We say that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) is true if the two sentences </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >are both true, and is false otherwise.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.1. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \2 + 2 = 4 and 2 + 3 = 5" is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.2. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \2 + 2 = 4 and </span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational" is false.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >1 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >(DISJUNCTION) We say that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >or </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) is true if at least one of two<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >sentences </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true, and is false otherwise.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.3. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \2 + 2 = 2 or 1 + 3 = 5" is false.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.4. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \2 + 2 = 4 or </span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational" is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Remark. </span><span style=";font-family:CMR10;font-size:10;" >To prove that a sentence </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >_</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true, we may assume that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is false and use this<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >to deduce that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true in this case. For if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is true, our argument is already<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >complete, never mind the truth or falsity of the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >(NEGATION) We say that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >(not </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >) is true if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is false, and is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >false if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.5. </span><span style=";font-family:CMR10;font-size:10;" >The negation of the sentence \2 + 2 = 4" is the sentence \2 + 2 </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= 4".<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.6. </span><span style=";font-family:CMR10;font-size:10;" >The negation of the sentence \</span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational" is the sentence \</span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is irrational".<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >(CONDITIONAL) We say that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >(if </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, then </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) is true if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >is false or if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true or both, and is false otherwise.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Remark. </span><span style=";font-family:CMR10;font-size:10;" >It is convenient to realize that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is false precisely when the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >true and the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is false. To understand this, note that if we draw a false conclusion from a true<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >assumption, then our argument must be faulty. On the other hand, if our assumption is false or if our<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >conclusion is true, then our argument may still be acceptable.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.7. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \if 2 + 2 = 2, then 1 + 3 = 5" is true, because the sentence \2 + 2 = 2"<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >is false.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.8. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \if 2 + 2 = 4, then </span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational" is false.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.9. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \if </span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational, then 2 + 2 = 4" is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >(DOUBLE CONDITIONAL) We say that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >if and only if </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) is true if<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >the two sentences </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >are both true or both false, and is false otherwise.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.10. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \2 + 2 = 4 if and only if </span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is irrational" is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.11. </span><span style=";font-family:CMR10;font-size:10;" >The sentence \2 + 2 </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= 4 if and only if </span><span style=";font-family:CMMI10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >is rational" is also true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >If we use the letter </span><span style=";font-family:CMMI10;font-size:10;" >T </span><span style=";font-family:CMR10;font-size:10;" >to denote \true" and the letter </span><span style=";font-family:CMMI10;font-size:10;" >F </span><span style=";font-family:CMR10;font-size:10;" >to denote \false", then the above _ve de_nitions<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >can be summarized in the following \truth table":<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >p q p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q p p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >T T T T F T T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >T F F T F F F<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >F T F T T T F<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >F F F F T T T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Remark. </span><span style=";font-family:CMR10;font-size:10;" >Note that in logic, \or" can mean \both". If you ask a logician whether he likes tea or co_ee,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >do not be surprised if he wants both!<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >2 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.1.12. </span><span style=";font-family:CMR10;font-size:10;" >The sentence (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) is true if exactly one of the two sentences </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >and is false otherwise; we have the following \truth table":<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >p q p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >T T T T F F<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >T F F T T T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >F T F T T T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >F F F F T F<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >1.2. Tautologies and Logical Equivalence<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >A tautology is a sentence which is true on logical ground only.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.1. </span><span style=";font-family:CMR10;font-size:10;" >The sentences (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) and (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >) are both tautologies.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >This enables us to generalize the de_nition of conjunction to more than two sentences, and write, for<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >example, </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >without causing any ambiguity.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.2. </span><span style=";font-family:CMR10;font-size:10;" >The sentences (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) and (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >) are both tautologies.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >This enables us to generalize the de_nition of disjunction to more than two sentences, and write, for<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >example, </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >without causing any ambiguity.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.3. </span><span style=";font-family:CMR10;font-size:10;" >The sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is a tautology.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.4. </span><span style=";font-family:CMR10;font-size:10;" >The sentence (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >) is a tautology.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.5. </span><span style=";font-family:CMR10;font-size:10;" >The sentence (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) is a tautology.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.6. </span><span style=";font-family:CMR10;font-size:10;" >The sentence (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)) is a tautology; we have the following \truth<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >table":<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >p q p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >))<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >T T T F F T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >T F F T T T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >F T F T T T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >F F T F F T<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >The following are tautologies which are commonly used. Let </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >denote sentences.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >DISTRIBUTIVE LAW. </span><span style=";font-family:CMTI10;font-size:10;" >The following sentences are tautologies:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(a) </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMTI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(b) </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMTI10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >DE MORGAN LAW. </span><span style=";font-family:CMTI10;font-size:10;" >The following sentences are tautologies:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(a) </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMTI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(b) </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMTI10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >INFERENCE LAW. </span><span style=";font-family:CMTI10;font-size:10;" >The following sentences are tautologies:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(a) </span><span style=";font-family:CMR10;font-size:10;" >(MODUS PONENS) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMTI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(b) </span><span style=";font-family:CMR10;font-size:10;" >(MODUS TOLLENS) ((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMTI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(c) </span><span style=";font-family:CMR10;font-size:10;" >(LAW OF SYLLOGISM) ((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMTI10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >3 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >These tautologies can all be demonstrated by truth tables. However, let us try to prove the _rst<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Distributive law here.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose _rst of all that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) is true. Then the two sentences </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >are both<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >true. Since the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >is true, at least one of the two sentences </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >is true. Without loss of<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >generality, assume that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true. Then the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >^</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true. It follows that the sentence<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose now that the sentence (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) is true. Then at least one of the two sentences (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >),<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >^</span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) is true. Without loss of generality, assume that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >^</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true. Then the two sentences<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >are both true. It follows that the sentence </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >is true, and so the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >It now follows that the two sentences </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >^</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_</span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) and (</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >^</span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >_</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMSY10;font-size:10;" >^</span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) are either both true or both false,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >as the truth of one implies the truth of the other. It follows that the double conditional (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) is a tautology.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >We say that two sentences </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >are logically equivalent if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is a<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >tautology.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.2.7. </span><span style=";font-family:CMR10;font-size:10;" >The sentences </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >are logically equivalent. The latter is known as the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >contrapositive of the former.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Remark. </span><span style=";font-family:CMR10;font-size:10;" >The sentences </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >are </span><span style=";font-family:CMBX10;font-size:10;" >not </span><span style=";font-family:CMR10;font-size:10;" >logically equivalent. The latter is known as the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >converse of the former.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >1.3. Sentential Functions and Sets<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >In many instances, we have sentences, such as \</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >is even", which contains one or more variables. We<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >shall call them sentential functions (or propositional functions).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Let us concentrate on our example \</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >is even". This sentence is true for certain values of </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, and is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >false for others. Various questions arise:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >What values of </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >do we permit?<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >Is the statement true for all such values of </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >in question?<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >Is the statement true for some such values of </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >in question?<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >To answer the _rst of these questions, we need the notion of a universe. We therefore need to consider<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >sets.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >We shall treat the word \set" as a word whose meaning everybody knows. Sometimes we use the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >synonyms \class" or \collection". However, note that in some books, these words may have di_erent<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >meanings!<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >The important thing about a set is what it contains. In other words, what are its members? Does it<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >have any? If </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >is a set and </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >is an element of </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >, we write </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >A set is usually described in one of the two following ways:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >By enumeration, </span><span style=";font-family:CMTI10;font-size:10;" >e.g. </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >3</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >denotes the set consisting of the numbers 1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >3 and nothing else;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >By a de_ning property (sentential function) </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >). Here it is important to de_ne a universe </span><span style=";font-family:CMMI10;font-size:10;" >U </span><span style=";font-family:CMR10;font-size:10;" >to<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >which all the </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >have to belong. We then write </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >U </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is true</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >or, simply,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >The set with no elements is called the empty set and denoted by </span><span style=";font-family:CMSY10;font-size:10;" >;</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >4 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.3.1. </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >3</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >4</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >5</span><span style=";font-family:CMMI10;font-size:10;" >; :::</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >is called the set of natural numbers.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.3.2. </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >: : : ;</span><span style=";font-family:";font-size:10;" >��</span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >;</span><span style=";font-family:";font-size:10;" >��</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >0</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >; : : :</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >is called the set of integers.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.3.3. </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:";font-size:10;" >��</span><span style=";font-family:CMSY10;font-size:10;" > </span><span style=";font-family:CMR10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" ><><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.3.4. </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:";font-size:10;" >��</span><span style=";font-family:CMSY10;font-size:10;" > </span><span style=";font-family:CMR10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" ><><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:";font-size:10;" >��</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >0</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.3.5. </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:";font-size:10;" >��</span><span style=";font-family:CMSY10;font-size:10;" > </span><span style=";font-family:CMR10;font-size:10;" >1 </span><span style=";font-family:CMMI10;font-size:10;" ><><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >;</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >1.4. Set Functions<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose that the sentential functions </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >), </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) are related to sets </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >with respect to a given universe,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >i.e. </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >. We de_ne<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >the intersection </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >the union </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >the complement </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >; and<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >the di_erence </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >n </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >The above are also sets. It is not di_cult to see that<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >or </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >62 </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >; and<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >n </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >62 </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >We say that the set </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >is a subset of the set </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >, denoted by </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >or by </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >, if every element of </span><span style=";font-family:CMMI10;font-size:10;" >P<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >is an element of </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >. In other words, if we have </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >with respect to some<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >universe </span><span style=";font-family:CMMI10;font-size:10;" >U</span><span style=";font-family:CMR10;font-size:10;" >, then we have </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >if and only if the sentence </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is true for all </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >U</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >We say that two sets </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >are equal, denoted by </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >, if they contain the same elements, </span><span style=";font-family:CMTI10;font-size:10;" >i.e.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >if each is a subset of the other, </span><span style=";font-family:CMTI10;font-size:10;" >i.e. </span><span style=";font-family:CMR10;font-size:10;" >if </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Furthermore, we say that </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >is a proper subset of </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >, denoted by </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >or by </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >, if </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >and<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >The following results on set functions can be deduced from their analogues in logic.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >DISTRIBUTIVE LAW. </span><span style=";font-family:CMTI10;font-size:10;" >If </span><span style=";font-family:CMMI10;font-size:10;" >P; Q;R </span><span style=";font-family:CMTI10;font-size:10;" >are sets, then<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(a) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) = (</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMTI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(b) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) = (</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMTI10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >DE MORGAN LAW. </span><span style=";font-family:CMTI10;font-size:10;" >If </span><span style=";font-family:CMMI10;font-size:10;" >P;Q </span><span style=";font-family:CMTI10;font-size:10;" >are sets, then with respect to a universe </span><span style=";font-family:CMMI10;font-size:10;" >U</span><span style=";font-family:CMTI10;font-size:10;" >,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(a) </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) = </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMTI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMTI10;font-size:10;" >(b) </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) = </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMTI10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >We now try to deduce the _rst Distributive law for set functions from the _rst Distributive law for<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >sentential functions.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose that the sentential functions </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >), </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >), </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) are related to sets </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >with respect to a<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >given universe, </span><span style=";font-family:CMTI10;font-size:10;" >i.e. </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >. Then<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) = </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMSY10;font-size:10;" >g<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >and<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) = </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: (</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMMI10;font-size:10;" >:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >5 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose that </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >). Then </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) is true. By the _rst Distributive law for<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >sentential functions, we have that<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >))) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)))<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >is a tautology. It follows that (</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) is true, so that </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >). This<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >gives<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMMI10;font-size:10;" >: </span><span style=";font-family:CMR10;font-size:10;" >(1)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose now that </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >). Then (</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) is true. It follows from the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >_rst Distributive law for sentential functions that </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)) is true, so that </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >This gives<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMMI10;font-size:10;" >: </span><span style=";font-family:CMR10;font-size:10;" >(2)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >The result now follows on combining (1) and (2).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >1.5. Quanti_er Logic<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Let us return to the example \</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >is even" at the beginning of Section 1.3.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose now that we restrict </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >to lie in the set </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >of all integers. Then the sentence \</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >is even" is only<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >true for some </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >in </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMR10;font-size:10;" >. It follows that the sentence \some </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >are even" is true, while the sentence \all<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >are even" is false.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >In general, consider a sentential function of the form </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >), where the variable </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >lies in some clearly<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >stated set. We can then consider the following two sentences:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ 8</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) (for all </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is true); and<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >_ 9</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) (for some </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is true).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Definition. </span><span style=";font-family:CMR10;font-size:10;" >The symbols </span><span style=";font-family:CMSY10;font-size:10;" >8 </span><span style=";font-family:CMR10;font-size:10;" >(for all) and </span><span style=";font-family:CMSY10;font-size:10;" >9 </span><span style=";font-family:CMR10;font-size:10;" >(for some) are called the universal quanti_er and the exis-<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >tential quanti_er respectively.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Note that the variable </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >is a \dummy variable". There is no di_erence between writing </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) or<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >writing </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR10;font-size:10;" >).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.5.1. </span><span style=";font-family:CMR10;font-size:10;" >(LAGRANGE'S THEOREM) Every natural number is the sum of the squares of four<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >integers. This can be written, in logical notation, as<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >n </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >a; b; c; d </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMMI10;font-size:10;" >; n </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >a</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >b</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >c</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >d</span><span style=";font-family:CMR7;font-size:7;" >2</span><span style=";font-family:CMMI10;font-size:10;" >:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.5.2. </span><span style=";font-family:CMR10;font-size:10;" >(GOLDBACH CONJECTURE) Every even natural number greater than 2 is the sum<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >of two primes. This can be written, in logical notation, as<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >n </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMSY10;font-size:10;" >n f</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >p; q </span><span style=";font-family:CMR10;font-size:10;" >prime</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >n </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >q:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >It is not yet known whether this is true or not. This is one of the greatest unsolved problems in<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >mathematics.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMBX10;font-size:10;" >1.6. Negation<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Our main concern is to develop a rule for negating sentences with quanti_ers. Let me start by saying<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >that you are all fools. Naturally, you will disagree, and some of you will complain. So it is natural to<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >suspect that the negation of the sentence </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is the sentence </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >6 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >There is another way to look at this. Let </span><span style=";font-family:CMMI10;font-size:10;" >U </span><span style=";font-family:CMR10;font-size:10;" >be the universe for all the </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >. Let </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >. Suppose<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >_rst of all that the sentence </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is true. Then </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >U</span><span style=";font-family:CMR10;font-size:10;" >, so </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >;</span><span style=";font-family:CMR10;font-size:10;" >. But </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >, so that if<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >the sentence </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) were true, then </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >;</span><span style=";font-family:CMR10;font-size:10;" >, a contradiction. On the other hand, suppose now that the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >sentence </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is false. Then </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >U</span><span style=";font-family:CMR10;font-size:10;" >, so that </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >;</span><span style=";font-family:CMR10;font-size:10;" >. It follows that the sentence </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Now let me moderate a bit and say that some of you are fools. You will still complain, so perhaps<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >none of you are fools. It is then natural to suspect that the negation of the sentence </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >) is the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >sentence </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR10;font-size:10;" >).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >To summarize, we simply \change the quanti_er to the other type and negate the sentential function".<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >Suppose now that we have something more complicated. Let us apply bit by bit our simple rule. For<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >example, the negation of<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >z; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >w; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y; z;w</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >z; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >w; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y; z;w</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMMI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >which is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >z; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >w; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y; z;w</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMMI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >which is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >z; </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >w; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y; z;w</span><span style=";font-family:CMR10;font-size:10;" >))</span><span style=";font-family:CMMI10;font-size:10;" >;<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >which is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >z; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >w; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y; z;w</span><span style=";font-family:CMR10;font-size:10;" >)</span><span style=";font-family:CMMI10;font-size:10;" >:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >It is clear that the rule is the following: Keep the variables in their original order. Then, alter all the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >quanti_ers. Finally, negate the sentential function.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Example 1.6.1. </span><span style=";font-family:CMR10;font-size:10;" >The negation of the Goldbach conjecture is, in logical notation,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >n </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMSY10;font-size:10;" >n f</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >p; q </span><span style=";font-family:CMR10;font-size:10;" >prime</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >n </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >q:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >In other words, there is an even natural number greater than 2 which is not the sum of two primes. In<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >summary, to disprove the Goldbach conjecture, we simply need one counterexample!<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >7 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMCSC10;font-size:10;" >Problems for Chapter 1<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >1. Using truth tables or otherwise, check that each of the following is a tautology:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) b) ((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >c) </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >) d) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >p<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >e) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >2. Decide (and justify) whether each of the following is a tautology:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)) b) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >))<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >c) ((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) d) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >s </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >t</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >e) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) f) </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >g) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >)) h) ((</span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >s</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >s</span><span style=";font-family:CMR10;font-size:10;" >)))<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >i) </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >s</span><span style=";font-family:CMR10;font-size:10;" >)) j) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >s</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >t </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMMI10;font-size:10;" >u</span><span style=";font-family:CMR10;font-size:10;" >)<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >k) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) l) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" ><span style=""> </span></span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" ><span style=""> </span></span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" ><span style=""> </span></span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >).<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >m) (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >s</span><span style=";font-family:CMR10;font-size:10;" >))) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >((</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >s</span><span style=";font-family:CMR10;font-size:10;" >))<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >3. For each of the following, decide whether the statement is true or false, and justify your assertion:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) If </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is true and </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is false, then </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >b) If </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMR10;font-size:10;" >is true, </span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMR10;font-size:10;" >is false and </span><span style=";font-family:CMMI10;font-size:10;" >r </span><span style=";font-family:CMR10;font-size:10;" >is false, then </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) is true.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >c) The sentence (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >$ </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >) is a tautology.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >d) The sentences </span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) and (</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >^ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >p </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >r</span><span style=";font-family:CMR10;font-size:10;" >) are logically equivalent.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >4. List the elements of each of the following sets:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >< </span><span style=";font-family:CMR10;font-size:10;" >45</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >b) </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMMI10;font-size:10;" >< </span><span style=";font-family:CMR10;font-size:10;" >45</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >c) </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ 2</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >= 0</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >d) </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ 4 = 6</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >e) </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >4 </span><span style=";font-family:CMR10;font-size:10;" >= 1</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >f) </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >4 </span><span style=";font-family:CMR10;font-size:10;" >= 1</span><span style=";font-family:CMSY10;font-size:10;" >g<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >5. How many elements are there in each of the following sets? Are the sets all di_erent?<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMSY10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >b) </span><span style=";font-family:CMSY10;font-size:10;" >f;g </span><span style=";font-family:CMR10;font-size:10;" >c) </span><span style=";font-family:CMSY10;font-size:10;" >ff;gg </span><span style=";font-family:CMR10;font-size:10;" >d) </span><span style=";font-family:CMSY10;font-size:10;" >f;</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMSY10;font-size:10;" >f;gg </span><span style=";font-family:CMR10;font-size:10;" >e) </span><span style=";font-family:CMSY10;font-size:10;" >f;</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMSY10;font-size:10;" >;g<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >6. Let </span><span style=";font-family:CMMI10;font-size:10;" >U </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >a; b; c; d</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >a; b</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >a; c; d</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >. Write down the elements of the following sets:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >b) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >c) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMR10;font-size:10;" >d) </span><span style=";font-family:CMMI10;font-size:10;" >Q<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >7. Let </span><span style=";font-family:CMMI10;font-size:10;" >U </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:MSBM10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x > </span><span style=";font-family:CMR10;font-size:10;" >0</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x > </span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMSY10;font-size:10;" >g </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >: </span><span style=";font-family:CMMI10;font-size:10;" >x < </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >. Find each of the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >following sets:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMR10;font-size:10;" >b) </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >c) </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >d) </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >B<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >e) </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >f) </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >g) </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMR10;font-size:10;" >h) </span><span style=";font-family:CMMI10;font-size:10;" >B<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >i) </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >j) </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >n </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMR10;font-size:10;" >k) </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMSY10;font-size:10;" >n </span><span style=";font-family:CMMI10;font-size:10;" >C<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >8. List all the subsets of the set </span><span style=";font-family:CMSY10;font-size:10;" >f</span><span style=";font-family:CMR10;font-size:10;" >1</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >2</span><span style=";font-family:CMMI10;font-size:10;" >; </span><span style=";font-family:CMR10;font-size:10;" >3</span><span style=";font-family:CMSY10;font-size:10;" >g</span><span style=";font-family:CMR10;font-size:10;" >. How many subsets are there?<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >9. </span><span style=";font-family:CMMI10;font-size:10;" >A</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >B</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >C</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >D </span><span style=";font-family:CMR10;font-size:10;" >are sets such that </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >D</span><span style=";font-family:CMR10;font-size:10;" >, and both </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >D </span><span style=";font-family:CMR10;font-size:10;" >are empty.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) Show by examples that </span><span style=";font-family:CMMI10;font-size:10;" >A </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >D </span><span style=";font-family:CMR10;font-size:10;" >can be empty.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >b) Show that if </span><span style=";font-family:CMMI10;font-size:10;" >C </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >A</span><span style=";font-family:CMR10;font-size:10;" >, then </span><span style=";font-family:CMMI10;font-size:10;" >B </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >D</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >10. Suppose that </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >R </span><span style=";font-family:CMR10;font-size:10;" >are subsets of </span><span style=";font-family:MSBM10;font-size:10;" >N</span><span style=";font-family:CMR10;font-size:10;" >. For each of the following, state whether or not the<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >statement is true, and justify your assertion by studying the analogous sentences in logic:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >) = (</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >Q</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >\ </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >[ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >). b) </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >if and only if </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >P</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >c) If </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >Q </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >, then </span><span style=";font-family:CMMI10;font-size:10;" >P </span><span style=";font-family:CMSY10;font-size:10;" >_ </span><span style=";font-family:CMMI10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" >8 of 9<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR7;font-size:7;" ><o:p> </o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >11. For each of the following sentences, write down the sentence in logical notation, negate the sentence,<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >and say whether the sentence or its negation is true:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) Given any integer, there is a larger integer.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >b) There is an integer greater than all other integers.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >c) Every even number is a sum of two odd numbers.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >d) Every odd number is a sum of two even numbers.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >e) The distance between any two complex numbers is positive.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >f) All natural numbers divisible by 2 and by 3 are divisible by 6.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >[</span><span style=";font-family:CMCSC10;font-size:10;" >Notation</span><span style=";font-family:CMR10;font-size:10;" >: Write </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >j </span><span style=";font-family:CMMI10;font-size:10;" >y </span><span style=";font-family:CMR10;font-size:10;" >if </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >divides </span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR10;font-size:10;" >.]<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >g) Every integer is a sum of the squares of two integers.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >h) There is no greatest natural number.<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >12. For each of the following sentences, express the sentence in words, negate the sentence, and say<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >whether the sentence or its negation is true:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >z </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >z</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >N </span><span style=";font-family:CMR10;font-size:10;" >b) </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >z </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >z</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR7;font-size:7;" >2<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >c) </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >Z</span><span style=";font-family:CMR10;font-size:10;" >, (</span><span style=";font-family:CMMI10;font-size:10;" >x > y</span><span style=";font-family:CMR10;font-size:10;" >) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMSY10;font-size:10;" >6</span><span style=";font-family:CMR10;font-size:10;" >= </span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR10;font-size:10;" >) d) </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >x; y; z </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >w </span><span style=";font-family:CMSY10;font-size:10;" >2 </span><span style=";font-family:MSBM10;font-size:10;" >R</span><span style=";font-family:CMR10;font-size:10;" >, </span><span style=";font-family:CMMI10;font-size:10;" >x</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >+ </span><span style=";font-family:CMMI10;font-size:10;" >z</span><span style=";font-family:CMR7;font-size:7;" >2 </span><span style=";font-family:CMR10;font-size:10;" >= 8</span><span style=";font-family:CMMI10;font-size:10;" >w<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >13. Let </span><span style=";font-family:CMMI10;font-size:10;" >p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y</span><span style=";font-family:CMR10;font-size:10;" >) be a sentential function with variables </span><span style=";font-family:CMMI10;font-size:10;" >x </span><span style=";font-family:CMR10;font-size:10;" >and </span><span style=";font-family:CMMI10;font-size:10;" >y</span><span style=";font-family:CMR10;font-size:10;" >. Discuss whether each of the following is<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >true on logical grounds only:<o:p></o:p></span></p> <p class="MsoNormal" style=""><span style=";font-family:CMR10;font-size:10;" >a) (</span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y</span><span style=";font-family:CMR10;font-size:10;" >)) b) (</span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; </span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y</span><span style=";font-family:CMR10;font-size:10;" >)) </span><span style=";font-family:CMSY10;font-size:10;" >! </span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMSY10;font-size:10;" >9</span><span style=";font-family:CMMI10;font-size:10;" >x; </span><span style=";font-family:CMSY10;font-size:10;" >8</span><span style=";font-family:CMMI10;font-size:10;" >y; p</span><span style=";font-family:CMR10;font-size:10;" >(</span><span style=";font-family:CMMI10;font-size:10;" >x; y</span><span style=";font-family:CMR10;font-size:10;" >))<o:p></o:p></span></p> <p class="MsoNormal"><span style=";font-family:CMR7;font-size:7;" >9 of 9</span><o:p></o:p></p> Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com2tag:blogger.com,1999:blog-7399621945608861143.post-42152492738746266342008-12-15T00:09:00.000-08:002008-12-23T01:55:02.553-08:00INTRODUCTION TO INEQUALITIES<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///D:%5CUSERPR%7E1%5Csshende%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///D:%5CUSERPR%7E1%5Csshende%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///D:%5CUSERPR%7E1%5Csshende%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:trackmoves/> <w:trackformatting/> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:donotpromoteqf/> <w:lidthemeother>EN-US</w:LidThemeOther> <w:lidthemeasian>JA</w:LidThemeAsian> <w:lidthemecomplexscript>X-NONE</w:LidThemeComplexScript> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> <w:splitpgbreakandparamark/> <w:dontvertaligncellwithsp/> <w:dontbreakconstrainedforcedtables/> <w:dontvertalignintxbx/> <w:word11kerningpairs/> <w:cachedcolbalance/> <w:usefelayout/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathpr> <m:mathfont val="Cambria Math"> <m:brkbin val="before"> <m:brkbinsub val="--"> <m:smallfrac val="off"> <m:dispdef/> <m:lmargin val="0"> <m:rmargin val="0"> <m:defjc val="centerGroup"> <m:wrapindent val="1440"> <m:intlim val="subSup"> <m:narylim val="undOvr"> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" defunhidewhenused="true" defsemihidden="true" defqformat="false" defpriority="99" latentstylecount="267"> <w:lsdexception locked="false" priority="0" semihidden="false" unhidewhenused="false" qformat="true" name="Normal"> <w:lsdexception locked="false" priority="9" semihidden="false" unhidewhenused="false" qformat="true" name="heading 1"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 2"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 3"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 4"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 5"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 6"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 7"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 8"> <w:lsdexception locked="false" priority="9" qformat="true" name="heading 9"> <w:lsdexception locked="false" priority="39" name="toc 1"> <w:lsdexception locked="false" priority="39" name="toc 2"> <w:lsdexception locked="false" priority="39" name="toc 3"> <w:lsdexception locked="false" priority="39" name="toc 4"> <w:lsdexception locked="false" priority="39" name="toc 5"> <w:lsdexception locked="false" priority="39" name="toc 6"> <w:lsdexception locked="false" priority="39" name="toc 7"> <w:lsdexception locked="false" priority="39" name="toc 8"> <w:lsdexception locked="false" priority="39" name="toc 9"> <w:lsdexception locked="false" priority="35" qformat="true" name="caption"> <w:lsdexception locked="false" priority="10" semihidden="false" unhidewhenused="false" qformat="true" name="Title"> <w:lsdexception locked="false" priority="1" name="Default Paragraph Font"> <w:lsdexception locked="false" priority="11" semihidden="false" unhidewhenused="false" qformat="true" name="Subtitle"> <w:lsdexception locked="false" priority="22" semihidden="false" unhidewhenused="false" qformat="true" name="Strong"> <w:lsdexception locked="false" priority="20" semihidden="false" unhidewhenused="false" qformat="true" name="Emphasis"> <w:lsdexception locked="false" priority="59" semihidden="false" unhidewhenused="false" name="Table Grid"> <w:lsdexception locked="false" unhidewhenused="false" name="Placeholder Text"> <w:lsdexception locked="false" priority="1" semihidden="false" unhidewhenused="false" qformat="true" name="No Spacing"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 1"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 1"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 1"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 1"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 1"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 1"> <w:lsdexception locked="false" unhidewhenused="false" name="Revision"> <w:lsdexception locked="false" priority="34" semihidden="false" unhidewhenused="false" qformat="true" name="List Paragraph"> <w:lsdexception locked="false" priority="29" semihidden="false" unhidewhenused="false" qformat="true" name="Quote"> <w:lsdexception locked="false" priority="30" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Quote"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 1"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 1"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 1"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 1"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 1"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 1"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 1"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 1"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 2"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 2"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 2"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 2"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 2"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 2"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 2"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 2"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 2"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 2"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 2"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 2"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 2"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 2"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 3"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 3"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 3"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 3"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 3"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 3"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 3"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 3"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 3"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 3"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 3"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 3"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 3"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 3"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 4"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 4"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 4"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 4"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 4"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 4"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 4"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 4"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 4"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 4"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 4"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 4"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 4"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 4"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 5"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 5"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 5"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 5"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 5"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 5"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 5"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 5"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 5"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 5"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 5"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 5"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 5"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 5"> <w:lsdexception locked="false" priority="60" semihidden="false" unhidewhenused="false" name="Light Shading Accent 6"> <w:lsdexception locked="false" priority="61" semihidden="false" unhidewhenused="false" name="Light List Accent 6"> <w:lsdexception locked="false" priority="62" semihidden="false" unhidewhenused="false" name="Light Grid Accent 6"> <w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 6"> <w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 6"> <w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 6"> <w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 6"> <w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 6"> <w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 6"> <w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 6"> <w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 6"> <w:lsdexception locked="false" priority="71" semihidden="false" unhidewhenused="false" name="Colorful Shading Accent 6"> <w:lsdexception locked="false" priority="72" semihidden="false" unhidewhenused="false" name="Colorful List Accent 6"> <w:lsdexception locked="false" priority="73" semihidden="false" unhidewhenused="false" name="Colorful Grid Accent 6"> <w:lsdexception locked="false" priority="19" semihidden="false" unhidewhenused="false" qformat="true" name="Subtle Emphasis"> <w:lsdexception locked="false" priority="21" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Emphasis"> <w:lsdexception locked="false" priority="31" semihidden="false" unhidewhenused="false" qformat="true" name="Subtle Reference"> <w:lsdexception locked="false" priority="32" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Reference"> <w:lsdexception locked="false" priority="33" semihidden="false" unhidewhenused="false" qformat="true" name="Book Title"> <w:lsdexception locked="false" priority="37" name="Bibliography"> <w:lsdexception locked="false" priority="39" qformat="true" name="TOC Heading"> </w:LatentStyles> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:SimSun; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:宋体; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-1610611985 1107304683 0 0 159 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} @font-face {font-family:"\@SimSun"; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:CMBX10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR8; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMCSC10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMMI10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMSY10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR7; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMTI10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:MSBM10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMMI7; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:MSAM10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMSY7; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR6; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMEX10; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} @font-face {font-family:CMR5; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:auto; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:SimSun; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:ZH-CN;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:SimSun; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:ZH-CN;} .MsoPapDefault {mso-style-type:export-only; margin-bottom:10.0pt; line-height:115%;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.0in 1.0in 1.0in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"MS Mincho"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;} </style> <![endif]--> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">
<br /><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMCSC10;">Abstract. </span><span style="font-size: 8pt; font-family: CMR8;">This is a somewhat modified version of the notes I had prepared for a lecture on inequalities<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">that formed part of a training camp organized by the Association of Mathematics Teachers of India for<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">preparation for the Indian National Mathematical Olympiad (INMO) for students from Tamil Nadu.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1. </span><span style="font-size: 10pt; font-family: CMCSC10;">Basic idea of inequalities<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1.1. </span><span style="font-size: 10pt; font-family: CMBX10;">What we need to prove. </span><span style="font-size: 10pt; font-family: CMR10;">An “inequation” is an expression of the form:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">where </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMR10;">is an expression in terms of certain variables. An “inequality”’ is an inequation that is<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">satisfied for all values of the variables (within a certain range).<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">For instance:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">+ 1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">and<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">are both inequations. Among these, the first inequation is true for </span><span style="font-size: 10pt; font-family: CMTI10;">all </span><span style="font-size: 10pt; font-family: CMR10;">real </span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 10pt; font-family: CMR10;">, while the second<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">inequation is true for all values of </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">within a certain range.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Thus, when we talk of an inequality, we have the following in mind:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The underlying </span><span style="font-size: 10pt; font-family: CMTI10;">inequation<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The </span><span style="font-size: 10pt; font-family: CMTI10;">range of values </span><span style="font-size: 10pt; font-family: CMR10;">over which the inequality is true<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">A </span><span style="font-size: 10pt; font-family: CMTI10;">strict </span><span style="font-size: 10pt; font-family: CMR10;">inequation is an inequation of the form:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">F > </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">where </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMR10;">is an expression in terms of the variables.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Given any inequation </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0 we can consider the corresponding strict inequation </span><span style="font-size: 10pt; font-family: CMMI10;">F > </span><span style="font-size: 10pt; font-family: CMR10;">0.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Thus, when studying an inequality, we are interested in:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The underlying </span><span style="font-size: 10pt; font-family: CMTI10;">inequation<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The </span><span style="font-size: 10pt; font-family: CMTI10;">range of values </span><span style="font-size: 10pt; font-family: CMR10;">over which the inequality is true<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The values for which exact equality holds<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Some other points to note:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">Any inequation of the form </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">G </span><span style="font-size: 10pt; font-family: CMR10;">where </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">G </span><span style="font-size: 10pt; font-family: CMR10;">are both expressions can be written in the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">standard form as </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMSY10;">−</span><span style="font-size: 10pt; font-family: CMMI10;">G </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0. The original inequation is true for precisely those values for which<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the standard form is true. The equality conditions are also the same.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">An inequation of the form </span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">G </span><span style="font-size: 10pt; font-family: CMR10;">can be expressed as </span><span style="font-size: 10pt; font-family: CMMI10;">G</span><span style="font-size: 10pt; font-family: CMSY10;">−</span><span style="font-size: 10pt; font-family: CMMI10;">F </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0. Again, the original inequation<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">is true for precisely those values for which the standard form is true. The equality conditions<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">are also the same.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">c Vipul Naik, B.Sc. (Hons) Math and C.S., Chennai Mathematical Institute.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1.2. </span><span style="font-size: 10pt; font-family: CMBX10;">No square is negative. </span><span style="font-size: 10pt; font-family: CMR10;">This basic inequality states:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The range is all </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMSY10;">2 </span><span style="font-size: 10pt; font-family: MSBM10;">R </span><span style="font-size: 10pt; font-family: CMR10;">and equality holds iff </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">= 0.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">This can be generalized to something of the form:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, x</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , x</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">))</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ (</span><span style="font-size: 10pt; font-family: CMMI10;">g</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, x</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , x</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">))</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The range is all </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMSY10;">2 </span><span style="font-size: 10pt; font-family: MSBM10;">R </span><span style="font-size: 10pt; font-family: CMR10;">and equality holds iff </span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, x</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , x</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) = </span><span style="font-size: 10pt; font-family: CMMI10;">g</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, x</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , x</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) = 0.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 1. </span><span style="font-size: 10pt; font-family: CMR10;">Prove that </span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">4 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">4 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0 for all real </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 10pt; font-family: CMR10;">, equality holding iff </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">= 0.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMTI10;">Proof. </span><span style="font-size: 10pt; font-family: CMR10;">We use:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">4 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">4 </span><span style="font-size: 10pt; font-family: CMR10;">= (</span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ (</span><span style="font-size: 10pt; font-family: CMMI10;">xy</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Thus, (</span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">) plays th role of </span><span style="font-size: 10pt; font-family: CMMI10;">f </span><span style="font-size: 10pt; font-family: CMR10;">above and </span><span style="font-size: 10pt; font-family: CMMI10;">xy </span><span style="font-size: 10pt; font-family: CMR10;">plays the role of </span><span style="font-size: 10pt; font-family: CMMI10;">g<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Clearly then, the left-hand-side is nonnegative, and is 0 if and only if </span><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">xy </span><span style="font-size: 10pt; font-family: CMR10;">= 0, thus forcing<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">= 0. </span><span style="font-size: 10pt; font-family: MSAM10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We can extend the idea to sums of more than two squares:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 2. </span><span style="font-size: 10pt; font-family: CMR10;">Prove that </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">ab </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">bc </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">ca </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0 with equality holding only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">= 0.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMTI10;">Proof. </span><span style="font-size: 10pt; font-family: CMR10;">The left-hand-side can be expressed as 1</span><span style="font-size: 10pt; font-family: CMMI10;">/</span><span style="font-size: 10pt; font-family: CMR10;">2(</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ (</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">). So it is nonnegative and<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">can be zero only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">= 0.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Alternatively, the left hand side can also be written as 1</span><span style="font-size: 10pt; font-family: CMMI10;">/</span><span style="font-size: 10pt; font-family: CMR10;">2((</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">+</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+(</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">+</span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+(</span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">+</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">) and is hence<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">nonnegative, taking the value 0 if and only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">= 0 </span><span style="font-size: 10pt; font-family: MSAM10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Another problem (for which I’m not writing the solution here):<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 3. </span><span style="font-size: 10pt; font-family: CMR10;">Prove that </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">ab </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">bc </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">ca</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0 with equality holding only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">It turns out that one of the solution techniques for the previous problem can be applied to this one.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1.3. </span><span style="font-size: 10pt; font-family: CMBX10;">Manipulating about the inequality symbol. </span><span style="font-size: 10pt; font-family: CMR10;">The following results are typically used for manipulating<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">inequalities:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">We can </span><span style="font-size: 10pt; font-family: CMTI10;">add </span><span style="font-size: 10pt; font-family: CMR10;">two inequalities. The greater side gets added to the greater side, the smaller side<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">to the smaller side. If either inequality is strict, the resultant inequality is again strict. More<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">generally, the set of values for which the resultant inequality becomes equality is the intersection<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">of the corresponding sets for each inequality.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">We can multiply both sides of an inequality by a positive number. In general, however, we cannot<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">multiply two inequalities.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2. </span><span style="font-size: 10pt; font-family: CMCSC10;">Mean inequalities<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2.1. </span><span style="font-size: 10pt; font-family: CMBX10;">Definition of means. </span><span style="font-size: 10pt; font-family: CMR10;">A </span><span style="font-size: 10pt; font-family: CMTI10;">mean </span><span style="font-size: 10pt; font-family: CMR10;">is a good notion of </span><span style="font-size: 10pt; font-family: CMTI10;">average </span><span style="font-size: 10pt; font-family: CMR10;">for a collection of numbers. A mean of<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">numbers is thus typically a function from </span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">-tuples of reals to reals, such that:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">If all the members of the tuple are equal, the mean should be equal to all of them. That is, if<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">. . . a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">then the mean of </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The mean is a symmetric function of all the elements of the tuple, that is, if the elements are<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">permuted, the value of the mean remains unchanged. That is, the mean of </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">same as the mean of </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(1)</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(2)</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 7pt; font-family: CMR7;">)</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The mean of a collection of positive numbers should be between the smallest number and the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">largest number. That is, if </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">, the mean lies between </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The mean is an increasing function in each of the arguments. That is, if </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMSY7;">0</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">, then the mean of<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">+1</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is less than or equal to the mean of </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMSY7;">0</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">+1</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We now define some typical notions of mean:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Definition. </span><span style="font-size: 10pt; font-family: CMR10;">(1) The </span><span style="font-size: 10pt; font-family: CMBX10;">arithmetic mean</span><span style="font-size: 6pt; font-family: CMR6;">(defined) </span><span style="font-size: 10pt; font-family: CMR10;">of </span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">real numbers </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is defined as:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . a</span><span style="font-size: 7pt; font-family: CMMI7;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The arithmetic mean is a well-defined notion for </span><span style="font-size: 10pt; font-family: CMTI10;">any </span><span style="font-size: 10pt; font-family: CMR10;">collection of real numbers (positive, negative<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">or zero).<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(2) The </span><span style="font-size: 10pt; font-family: CMBX10;">geometric mean</span><span style="font-size: 6pt; font-family: CMR6;">(defined) </span><span style="font-size: 10pt; font-family: CMR10;">of </span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">positive real numbers </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is defined as<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">. . . a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 7pt; font-family: CMMI7;">/n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The geometric mean is defined only for </span><span style="font-size: 10pt; font-family: CMTI10;">positive </span><span style="font-size: 10pt; font-family: CMR10;">numbers.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(3) The </span><span style="font-size: 10pt; font-family: CMBX10;">quadratic mean</span><span style="font-size: 6pt; font-family: CMR6;">(defined) </span><span style="font-size: 10pt; font-family: CMR10;">or the root-mean-square of </span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">real numbers </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">defined as:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">21<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">22<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(4) The </span><span style="font-size: 10pt; font-family: CMBX10;">harmonic mean</span><span style="font-size: 6pt; font-family: CMR6;">(defined) </span><span style="font-size: 10pt; font-family: CMR10;">of </span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">nonzero real numbers </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">is defined as:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">For two positive reals </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">, these boil down to the formulas:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Name of the mean Value<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Arithmetic mean </span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">a</span><span style="font-size: 7pt; font-family: CMR7;">+</span><span style="font-size: 7pt; font-family: CMMI7;">b</span><span style="font-size: 7pt; font-family: CMR7;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Geometric mean </span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">ab<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Quadratic mean<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">q<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">a</span><span style="font-size: 5pt; font-family: CMR5;">2</span><span style="font-size: 7pt; font-family: CMR7;">+</span><span style="font-size: 7pt; font-family: CMMI7;">b</span><span style="font-size: 5pt; font-family: CMR5;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Harmonic mean </span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 7pt; font-family: CMMI7;">ab<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">a</span><span style="font-size: 7pt; font-family: CMR7;">+</span><span style="font-size: 7pt; font-family: CMMI7;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2.2. </span><span style="font-size: 10pt; font-family: CMBX10;">Inequalities for two variables.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Claim. </span><span style="font-size: 10pt; font-family: CMR10;">For positive reals </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">, Q.M. </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">A.M. </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">G.M. </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">H.M.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMTI10;">Proof. </span><span style="font-size: 10pt; font-family: CMR10;">We prove Q.M. </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">A.M. The remaining proofs follow along similar lines:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">What we would like to show is that, for all reals </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Since the left side is nonnegative, it suffices to show that the </span><span style="font-size: 10pt; font-family: CMTI10;">square </span><span style="font-size: 10pt; font-family: CMR10;">of the left side is greater than or<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">equal to the square of the right side. That is, we need to show that:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">But the latter rearranges to (</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0. This tells us that the inequality is valid for all real </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">with equality holding iff </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">. </span><span style="font-size: 10pt; font-family: MSAM10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">3<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Let’s look at the pattern. The Q.M. is essentially obtained by taking the arithmetic mean of squares<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">and then taking squareroot. The A.M. is obtained by taking the arithmetic mean of first powers and<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">then taking the first root. The H.M. is obtained by taking the arithmetic mean of inverses and then<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">taking the inverse. This suggests a general definition:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a, b</span><span style="font-size: 10pt; font-family: CMR10;">) =<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">r </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 7pt; font-family: CMMI7;">/r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Then the quadratic mean is </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">, the arithmetic mean is </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">, and the harmonic mean is </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">By this definition, </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">0 </span><span style="font-size: 10pt; font-family: CMR10;">does not make sense. But it turns out that, through a suitable limit argument,<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">we can take </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">0 </span><span style="font-size: 10pt; font-family: CMR10;">as the geometric mean. In that case, we have:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMR7;">0 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We also know that:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0 </span><span style="font-size: 10pt; font-family: CMSY10;">_ −</span><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Does this suggest something?<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2.3. </span><span style="font-size: 10pt; font-family: CMBX10;">The mean inequalities: an explanation. </span><span style="font-size: 10pt; font-family: CMR10;">Let </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">be positive reals. What can we say about<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the behaviour of </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a, b</span><span style="font-size: 10pt; font-family: CMR10;">) as </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMR10;">varies from </span><span style="font-size: 10pt; font-family: CMSY10;">−1 </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMSY10;">1</span><span style="font-size: 10pt; font-family: CMR10;">. It turns out that as </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMSY10;">! −1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r </span><span style="font-size: 10pt; font-family: CMR10;">approaches<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">min</span><span style="font-size: 10pt; font-family: CMSY10;">{</span><span style="font-size: 10pt; font-family: CMMI10;">a, b</span><span style="font-size: 10pt; font-family: CMSY10;">}</span><span style="font-size: 10pt; font-family: CMR10;">, and as </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMSY10;">! 1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r </span><span style="font-size: 10pt; font-family: CMSY10;">! </span><span style="font-size: 10pt; font-family: CMR10;">max</span><span style="font-size: 10pt; font-family: CMSY10;">{</span><span style="font-size: 10pt; font-family: CMMI10;">a, b</span><span style="font-size: 10pt; font-family: CMSY10;">}</span><span style="font-size: 10pt; font-family: CMR10;">. Thus, as </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMR10;">steadily increases, </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a, b</span><span style="font-size: 10pt; font-family: CMR10;">) steadily goes from<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the minimum to the maximum.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The explanation for this can be sought by viewing the </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMR10;">as a kind of </span><span style="font-size: 10pt; font-family: CMTI10;">weighting </span><span style="font-size: 10pt; font-family: CMR10;">of </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">. The greater<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the value of </span><span style="font-size: 10pt; font-family: CMMI10;">r</span><span style="font-size: 10pt; font-family: CMR10;">, the greater the </span><span style="font-size: 10pt; font-family: CMTI10;">dominance </span><span style="font-size: 10pt; font-family: CMR10;">of the bigger term, and hence, the greater the mean is to the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">bigger term.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2.4. </span><span style="font-size: 10pt; font-family: CMBX10;">The mean inequalities for many variables. </span><span style="font-size: 10pt; font-family: CMR10;">The same phenomena which we observe for two<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">variables also generalize to more than two variables. We define:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) =<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . a</span><span style="font-size: 7pt; font-family: CMMI7;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 7pt; font-family: CMMI7;">/r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Again, as </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMSY10;">! −1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r </span><span style="font-size: 10pt; font-family: CMR10;">approaches the minimum of the </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">s, and as </span><span style="font-size: 10pt; font-family: CMMI10;">r </span><span style="font-size: 10pt; font-family: CMSY10;">! 1</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">M</span><span style="font-size: 7pt; font-family: CMMI7;">r </span><span style="font-size: 10pt; font-family: CMR10;">approaches the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">maximum of the </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">s.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3. </span><span style="font-size: 10pt; font-family: CMCSC10;">Cauchy-Schwarz inequality<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3.1. </span><span style="font-size: 10pt; font-family: CMBX10;">Statement. </span><span style="font-size: 10pt; font-family: CMR10;">Let (</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) and (</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , b</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) be two </span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">-tuples of real numbers. Then:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">)(<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">(<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">))</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">With equality holding if and only if one of the tuples is zero or if </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">_a</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">for some fixed </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">independent<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">of </span><span style="font-size: 10pt; font-family: CMMI10;">i </span><span style="font-size: 10pt; font-family: CMR10;">(that is, the tuple of </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">s is a scalar multiple of the tuple of </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">s).<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3.2. </span><span style="font-size: 10pt; font-family: CMBX10;">Vector interpretation. </span><span style="font-size: 10pt; font-family: CMR10;">The vector interpretation of Cauchy Schwarz inequality looks at both<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= (</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">,ma</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) and </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= (</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , b</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) as vectors in </span><span style="font-size: 10pt; font-family: MSBM10;">R</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">. Then, the left-hand-side is:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">where </span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMSY10;">| </span><span style="font-size: 10pt; font-family: CMR10;">denotes the magnitude or length of the vector </span><span style="font-size: 10pt; font-family: CMMI10;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The right-hand-side is the square of the dot product of the vectors, which is the same as:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a.b</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMSY10;">|</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">cos</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">where </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">is the angle between the vectors. Since cos</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">1 and quality holds if and only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">are<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">collinear, we get a </span><span style="font-size: 10pt; font-family: CMTI10;">geometric </span><span style="font-size: 10pt; font-family: CMR10;">proof of Cauchy-Schwarz inequality.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">4<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3.3. </span><span style="font-size: 10pt; font-family: CMBX10;">A trigonometric problem. </span><span style="font-size: 10pt; font-family: CMR10;">Consider the following problem:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 4. </span><span style="font-size: 10pt; font-family: CMR10;">Maximize<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">as a function of </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">where </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">are fixed reals (and not both zero).<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The idea is to view this as a </span><span style="font-size: 10pt; font-family: CMTI10;">dot product </span><span style="font-size: 10pt; font-family: CMR10;">of vectors (</span><span style="font-size: 10pt; font-family: CMMI10;">a, b</span><span style="font-size: 10pt; font-family: CMR10;">) and (cos </span><span style="font-size: 10pt; font-family: CMMI10;">_, </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">). We have:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">)(cos</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ sin</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Since cos</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ sin</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">= 1, we obtain:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">p<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">A necessary and sufficient condition for the magnitude of the left-hand side to be </span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">is that<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a/ </span><span style="font-size: 10pt; font-family: CMR10;">cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b/ </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">, giving tan </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b/a</span><span style="font-size: 10pt; font-family: CMR10;">. Among the two possible values for the pair (cos </span><span style="font-size: 10pt; font-family: CMMI10;">_, </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">) we must<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">pick the one making </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">sin </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">positive.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3.4. </span><span style="font-size: 10pt; font-family: CMBX10;">A geometric problem. </span><span style="font-size: 10pt; font-family: CMR10;">Consider the following problem:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 5. </span><span style="font-size: 10pt; font-family: CMR10;">Let </span><span style="font-size: 10pt; font-family: CMMI10;">A </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">be two points in a plane at distance 1. Find the maximum length of a path<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">from </span><span style="font-size: 10pt; font-family: CMMI10;">A </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">B</span><span style="font-size: 10pt; font-family: CMR10;">, comprising at most </span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">line segments, with the property that at every stage, the distance<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">from </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">is reducing.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The answer is </span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMTI10;">Proof. </span><span style="font-size: 10pt; font-family: CMR10;">The idea of the proof is to use induction on </span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">. Let </span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">) denote the maximum value for a given </span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We observe that any such optimal path is </span><span style="font-size: 10pt; font-family: CMTI10;">memoryless </span><span style="font-size: 10pt; font-family: CMR10;">in the following sense:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Suppose </span><span style="font-size: 10pt; font-family: CMMI10;"><span style=""> </span></span><span style="font-size: 10pt; font-family: CMR10;">is a path from </span><span style="font-size: 10pt; font-family: CMMI10;">A </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">comprising at most </span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMR10;">line segments, and suppose that the first line<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">segment of </span><span style="font-size: 10pt; font-family: CMMI10;"><span style=""> </span></span><span style="font-size: 10pt; font-family: CMR10;">ends at a point </span><span style="font-size: 10pt; font-family: CMMI10;">P</span><span style="font-size: 10pt; font-family: CMR10;">. Now, the part from </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">must be composed of (</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1) line segments<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">with the property that at every stage, the distance from </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">is reducing.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Now, whatever path we choose, we could replace it by a path of maximum length from </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">B<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">comprising (</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1) line segments and with the property that distance from </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">is reducing. Since the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">original thing was </span><span style="font-size: 10pt; font-family: CMTI10;">longest</span><span style="font-size: 10pt; font-family: CMR10;">, we conclude that the part from </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">B </span><span style="font-size: 10pt; font-family: CMR10;">must also be the </span><span style="font-size: 10pt; font-family: CMTI10;">longest </span><span style="font-size: 10pt; font-family: CMR10;">one.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Now what is the longest possible path of (</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1) line segments from </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">B</span><span style="font-size: 10pt; font-family: CMR10;">? Since lengths scale, it<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">is the length </span><span style="font-size: 10pt; font-family: CMMI10;">PB </span><span style="font-size: 10pt; font-family: CMR10;">times the value </span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1). We thus get:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">length of</span><span style="font-size: 10pt; font-family: CMMI10;"> </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">AP </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">PBf</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Thus the maximum of the possible lengths of </span><span style="font-size: 10pt; font-family: CMMI10;"><span style=""> </span></span><span style="font-size: 10pt; font-family: CMR10;">is the maximum over all </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">of the above expression.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Now, from the fact that along the path </span><span style="font-size: 10pt; font-family: CMMI10;">AP</span><span style="font-size: 10pt; font-family: CMR10;">, the distance from </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">is steadily reducing, we obtain that<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the angle </span><span style="font-size: 10pt; font-family: MSAM10;">\</span><span style="font-size: 10pt; font-family: CMMI10;">APB </span><span style="font-size: 10pt; font-family: CMR10;">is either obtuse or right. Thus, in particular, for any given length </span><span style="font-size: 10pt; font-family: CMMI10;">AP</span><span style="font-size: 10pt; font-family: CMR10;">, we have:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">PB </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">p<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">AP</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">If equality does not hold, we could replace </span><span style="font-size: 10pt; font-family: CMMI10;">P </span><span style="font-size: 10pt; font-family: CMR10;">by another point </span><span style="font-size: 10pt; font-family: CMMI10;">Q </span><span style="font-size: 10pt; font-family: CMR10;">such that </span><span style="font-size: 10pt; font-family: CMMI10;">AQ </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">AP </span><span style="font-size: 10pt; font-family: CMR10;">and such that<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: MSAM10;">\</span><span style="font-size: 10pt; font-family: CMMI10;">AQB </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">_/</span><span style="font-size: 10pt; font-family: CMR10;">2. Then, </span><span style="font-size: 10pt; font-family: CMMI10;">QB </span><span style="font-size: 10pt; font-family: CMR10;">would be greater than </span><span style="font-size: 10pt; font-family: CMMI10;">PB</span><span style="font-size: 10pt; font-family: CMR10;">, and hence, the length of the longest path would<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">increase. Hence, we conclude that equality does indeed hold for the longest path, viz </span><span style="font-size: 10pt; font-family: MSAM10;">\</span><span style="font-size: 10pt; font-family: CMMI10;">APB </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">_/</span><span style="font-size: 10pt; font-family: CMR10;">2.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Let </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">be </span><span style="font-size: 10pt; font-family: MSAM10;">\</span><span style="font-size: 10pt; font-family: CMMI10;">BAP</span><span style="font-size: 10pt; font-family: CMR10;">. Then </span><span style="font-size: 10pt; font-family: CMMI10;">AP </span><span style="font-size: 10pt; font-family: CMR10;">= cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">PB </span><span style="font-size: 10pt; font-family: CMR10;">= sin </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">. We thus get:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">length of </span><span style="font-size: 10pt; font-family: CMMI10;"><span style=""> </span></span><span style="font-size: 10pt; font-family: CMR10;">= max<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">cos </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1) sin </span><span style="font-size: 10pt; font-family: CMMI10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Thus, applying the result of the previous problem:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">) =<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">p<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1 + (</span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">1))</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Since </span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(1) = 1 (clearly) we get </span><span style="font-size: 10pt; font-family: CMMI10;">f</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">) = </span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">. </span><span style="font-size: 10pt; font-family: MSAM10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">5<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4. </span><span style="font-size: 10pt; font-family: CMCSC10;">Rearrangement and Chebyshev inequality<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4.1. </span><span style="font-size: 10pt; font-family: CMBX10;">Rearrangement inequality: statement. </span><span style="font-size: 10pt; font-family: CMR10;">Let (</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) and (</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , b</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) be two </span><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">-tuples<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">of real numbers such that </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">n </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">. . . b</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">. Let </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">be a permutation of the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">numbers 1</span><span style="font-size: 10pt; font-family: CMMI10;">, </span><span style="font-size: 10pt; font-family: CMR10;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , n</span><span style="font-size: 10pt; font-family: CMR10;">. Then:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">In other words, the sum of pairwise products is maximum if we pair the </span><span style="font-size: 10pt; font-family: CMTI10;">largest </span><span style="font-size: 10pt; font-family: CMR10;">with the largest, the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">second largest with the second largest, and so on.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Equality holds if and only if, for each </span><span style="font-size: 10pt; font-family: CMMI10;">i</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">) </span><span style="font-size: 10pt; font-family: CMR10;">or </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">)</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Further:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 7pt; font-family: CMR7;">+1</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">In other words, the sum of pairwise products is minimum if we pair the largest with the smallest, the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">second largest with the second smallest, and so on.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4.2. </span><span style="font-size: 10pt; font-family: CMBX10;">Idea behind the inequality. </span><span style="font-size: 10pt; font-family: CMR10;">Think of it as a resource allocation problem. For instance, suppose<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">a thief has 3 bags and 3 kinds of coins (gold, silver, copper) to pack in them, and she must pack a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">different kind of coin in each bag. Assume further that the coins are available in unlimited quantities.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Then, in order to maximize her loot, she will put the gold coins in the biggest bag, the silver coins in<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the second biggest bag, and the copper coins in the third biggest bag.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The idea is: send the most to the best. Such an allocation principle is often called a </span><span style="font-size: 10pt; font-family: CMTI10;">greedy </span><span style="font-size: 10pt; font-family: CMR10;">allocation<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">principle.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The Rearrangement inequality is best proved for two elements, and then extended by induction. Let<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">. Then we have:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">)(</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Manipulating this gives us:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The rearrangement inequality thus illustrates the general statement the principles of optimization and<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">equality are often at crossroads.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">To use this to prove the result globally, we start with the expression<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">) </span><span style="font-size: 10pt; font-family: CMR10;">and locate indices </span><span style="font-size: 10pt; font-family: CMMI10;">i, j<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">for which </span><span style="font-size: 10pt; font-family: CMMI10;">i <><span style="font-size: 10pt; font-family: CMR10;">but </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">i</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMMI10;">> _</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">j</span><span style="font-size: 10pt; font-family: CMR10;">). We then change the permutation to one sending </span><span style="font-size: 10pt; font-family: CMMI10;">i </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">j</span><span style="font-size: 10pt; font-family: CMR10;">) and </span><span style="font-size: 10pt; font-family: CMMI10;">j </span><span style="font-size: 10pt; font-family: CMR10;">to </span><span style="font-size: 10pt; font-family: CMMI10;">_</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">i</span><span style="font-size: 10pt; font-family: CMR10;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(and having the same effect as </span><span style="font-size: 10pt; font-family: CMMI10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">on the others). This local change increases the value of the expression<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">and hence it is clearly not the optimum value.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Note here that equality holds only if </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">j </span><span style="font-size: 10pt; font-family: CMR10;">or </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 7pt; font-family: CMR7;">) </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">_</span><span style="font-size: 7pt; font-family: CMR7;">(</span><span style="font-size: 7pt; font-family: CMMI7;">j</span><span style="font-size: 7pt; font-family: CMR7;">)</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4.3. </span><span style="font-size: 10pt; font-family: CMBX10;">An application of rearrangement. </span><span style="font-size: 10pt; font-family: CMR10;">Consider the following problem I had mentioned earlier:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Prove that </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 7pt; font-family: CMR7;">2 </span><span style="font-size: 10pt; font-family: CMSY10;">− </span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">ab </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">bc </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">ca</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMR10;">0 with equality holding only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">This problem can also be solved using the rearrangement inequality. First observe that since the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">expression is symmetric in </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">, we can assume without loss of generality that </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Consider the triple (</span><span style="font-size: 10pt; font-family: CMMI10;">a, b, c</span><span style="font-size: 10pt; font-family: CMR10;">). This is an ordered triple with the property that the elements are in nonincreasing<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">order. Then (</span><span style="font-size: 10pt; font-family: CMMI10;">b, c, a</span><span style="font-size: 10pt; font-family: CMR10;">) is a permutation of this expression. Thus, by rearrangement inequality:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">aa </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">bb </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">cc </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">ab </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">bc </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">ca<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Which gives us what we want.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Also note that in this case, equality holds if and only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4.4. </span><span style="font-size: 10pt; font-family: CMBX10;">Chebyshev inequality. </span><span style="font-size: 10pt; font-family: CMR10;">Chebyshev inequality says that sending the most to the best is better<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">than giving the average to the average. More formally, if (</span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, a</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , a</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) and (</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, b</span><span style="font-size: 7pt; font-family: CMR7;">2</span><span style="font-size: 10pt; font-family: CMMI10;">, . . . , b</span><span style="font-size: 7pt; font-family: CMMI7;">n</span><span style="font-size: 10pt; font-family: CMR10;">) are two<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n</span><span style="font-size: 10pt; font-family: CMR10;">-tuples of decreasing reals:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">X<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">n<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Where equality holds iff either all the </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">s are equal or all the </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">s are equal.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">6<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">4.5. </span><span style="font-size: 10pt; font-family: CMBX10;">Fundamental difference between Chebyshev and Cauchy-Schwarz. </span><span style="font-size: 10pt; font-family: CMR10;">Both the Chebyshev<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">and the Cauchy-Schwarz inequalities are similar in the following sense:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">They are both true for all reals<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">They both provide bounds of<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">But they are different in the following ways:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">In Chebyshev, it is important to order the </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">s and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">s in descending order, whereas Cauchy-<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Schwarz is applicable for any ordering<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">Chebyshev gives a bound in terms of<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">and<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">while Cauchy-Schwarz gives a bound in<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">terms of the sums of their squares.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">Chebyshev provides a </span><span style="font-size: 10pt; font-family: CMTI10;">lower bound </span><span style="font-size: 10pt; font-family: CMR10;">on<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">while Cauchy-Schwarz provides an upper bound<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">• </span><span style="font-size: 10pt; font-family: CMR10;">The equality case is different in both. In Chebyshev, equality holds if all the elements in one of<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">the tuples are equal. In Cauchy-Schwarz, equality holds if the two tuples are scalar multiples of<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">one another.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">A word of caution, though, when deciding whether to apply Chebyshev or Cauchy-Schwarz. Just<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">because the inequality seems to require a lower bound on<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">P<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMMI10;">F</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">G</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">, does not mean that Chebyshev is the<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">one to be used. In fact, we could still use Cauchy-Schwarz by taking </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">F</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMMI10;">G</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMMI7;">i </span><span style="font-size: 10pt; font-family: CMR10;">to be 1</span><span style="font-size: 10pt; font-family: CMMI10;">/F</span><span style="font-size: 7pt; font-family: CMMI7;">i</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">5. </span><span style="font-size: 10pt; font-family: CMCSC10;">Nesbitt’s inequality<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">5.1. </span><span style="font-size: 10pt; font-family: CMBX10;">Statement of the inequality.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 6 </span><span style="font-size: 10pt; font-family: CMR10;">(Nesbitt’s inequality)</span><span style="font-size: 10pt; font-family: CMBX10;">. </span><span style="font-size: 10pt; font-family: CMR10;">For positive </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">, prove that:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">with equality holding if and only if </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">5.2. </span><span style="font-size: 10pt; font-family: CMBX10;">Applying Cauchy-Schwarz (direct application fails). </span><span style="font-size: 10pt; font-family: CMR10;">To apply Cauchy-Schwarz we need to<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">put the terms </span><span style="font-size: 7pt; font-family: CMMI7;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">b</span><span style="font-size: 7pt; font-family: CMR7;">+</span><span style="font-size: 7pt; font-family: CMMI7;">c </span><span style="font-size: 10pt; font-family: CMR10;">and its analogues on the </span><span style="font-size: 10pt; font-family: CMTI10;">left </span><span style="font-size: 10pt; font-family: CMR10;">side, which means we should view each of them as a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">square. Their squareroots are<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">q<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">b</span><span style="font-size: 7pt; font-family: CMR7;">+</span><span style="font-size: 7pt; font-family: CMMI7;">c </span><span style="font-size: 10pt; font-family: CMR10;">and its analogues. Thus, one tuple is:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;"><span style=""> </span>r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">,<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">,<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">r<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">c<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">!<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We would like the other tuple to be </span><span style="font-size: 10pt; font-family: "Arial","sans-serif";">��</span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c,</span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a,</span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">something that cancels the denominator. A natural choice is<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">. Unfortunately, this fails to yield the answer, because the expression that we<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">get is upper-bounded, rather than lower-bounded, in the case of equality.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">5.3. </span><span style="font-size: 10pt; font-family: CMBX10;">Applying Chebyshev. </span><span style="font-size: 10pt; font-family: CMR10;">Consider the tuples (</span><span style="font-size: 10pt; font-family: CMMI10;">a, b, c</span><span style="font-size: 10pt; font-family: CMR10;">) and ((</span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, </span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMMI10;">, </span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">). We first<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">need to determine whether they are arranged in the same order. Assume without loss of generality that<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">. Then </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMSY10;">_ </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">, and taking inverses, we obtain that the second tuple also has its<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">coordinates in descending order.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We are thus in a position to apply Chebyshev’s and obtain that the give expression is at least:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)((</span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">+ (</span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1 </span><span style="font-size: 10pt; font-family: CMR10;">+ (</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">)</span><span style="font-size: 7pt; font-family: CMSY7;">−</span><span style="font-size: 7pt; font-family: CMR7;">1</span><span style="font-size: 10pt; font-family: CMR10;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">Now using A.M.-H.M. inequality for the quantities (</span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">), (</span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">) and (</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">), we get the required<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">result.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">5.4. </span><span style="font-size: 10pt; font-family: CMBX10;">A short proof. </span><span style="font-size: 10pt; font-family: CMR10;">Another way of proving the result is to add and subtract 3, thus writing it as:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">And now apply the A.M.-H.M. inequality.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">7<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">6. </span><span style="font-size: 10pt; font-family: CMCSC10;">A past IMO problem<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">6.1. </span><span style="font-size: 10pt; font-family: CMBX10;">The problem statement.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMBX10;">Problem 7 </span><span style="font-size: 10pt; font-family: CMR10;">(IMO 1995)</span><span style="font-size: 10pt; font-family: CMBX10;">. </span><span style="font-size: 10pt; font-family: CMR10;">Prove that if </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">are positive reals such that </span><span style="font-size: 10pt; font-family: CMMI10;">abc </span><span style="font-size: 10pt; font-family: CMR10;">= 1, then:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 10pt; font-family: CMR10;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">a</span><span style="font-size: 10pt; font-family: CMR10;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">1<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">c</span><span style="font-size: 7pt; font-family: CMR7;">3</span><span style="font-size: 10pt; font-family: CMR10;">(</span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">b</span><span style="font-size: 10pt; font-family: CMR10;">) </span><span style="font-size: 10pt; font-family: CMSY10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">3<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">The first trick is to put </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">= 1</span><span style="font-size: 10pt; font-family: CMMI10;">/a</span><span style="font-size: 10pt; font-family: CMR10;">, </span><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">= 1</span><span style="font-size: 10pt; font-family: CMMI10;">/b </span><span style="font-size: 10pt; font-family: CMR10;">and </span><span style="font-size: 10pt; font-family: CMMI10;">z </span><span style="font-size: 10pt; font-family: CMR10;">= 1</span><span style="font-size: 10pt; font-family: CMMI10;">/c</span><span style="font-size: 10pt; font-family: CMR10;">. The left-hand side becomes:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">z<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">y</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">z </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">x<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">z</span><span style="font-size: 7pt; font-family: CMR7;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">y<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">6.2. </span><span style="font-size: 10pt; font-family: CMBX10;">Cauchy-Schwarz. </span><span style="font-size: 10pt; font-family: CMR10;">After this point, the first possibility to consider is Cauchy-Schwarz. Since we<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">want to lower-bound the sum here, we must view </span><span style="font-size: 7pt; font-family: CMMI7;">x</span><span style="font-size: 5pt; font-family: CMR5;">2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMMI7;">y</span><span style="font-size: 7pt; font-family: CMR7;">+</span><span style="font-size: 7pt; font-family: CMMI7;">z </span><span style="font-size: 10pt; font-family: CMR10;">and its analogues as squares of a tuple. The other<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">tuple is obtained by cancelling denominators from third tuple. We thus have tuples:<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">x<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">z<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">,<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">y<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">z </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">x<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">,<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">z<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">y<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">and<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: "Arial","sans-serif";">��</span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">z,</span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">z </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">x,</span><span style="font-size: 10pt; font-family: CMSY10;">p</span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">+ </span><span style="font-size: 10pt; font-family: CMMI10;">y<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMEX10;">_<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">We apply Cauchy-Schwarz to these tuples, and then use A.M.-G.M. inequality and the fact that<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMMI10;">xyz </span><span style="font-size: 10pt; font-family: CMR10;">= 1.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">If we keep track of the inequality constraints at each step, we obtain that equality holds if and only<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMR10;">if </span><span style="font-size: 10pt; font-family: CMMI10;">x </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">y </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">z </span><span style="font-size: 10pt; font-family: CMR10;">= 1, and hence </span><span style="font-size: 10pt; font-family: CMMI10;">a </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">b </span><span style="font-size: 10pt; font-family: CMR10;">= </span><span style="font-size: 10pt; font-family: CMMI10;">c </span><span style="font-size: 10pt; font-family: CMR10;">= 1.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 7pt; font-family: CMR7;">8<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 10pt; font-family: CMCSC10;">Index<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">arithmetic mean, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">Cauchy-Schwarz inequality, 3<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">Chebyshev inequality, 5<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">geometric mean, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">harmonic mean, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">mean<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">arithmetic, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">geometric, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">harmonic, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">quadratic, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">quadratic mean, 2<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal;"><span style="font-size: 8pt; font-family: CMR8;">rearrangement inequality, 4<o:p></o:p></span></p> <p class="MsoNormal"><span style="font-size: 7pt; line-height: 115%; font-family: CMR7;">9</span></p> Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-19020538633974029772008-08-12T04:36:00.001-07:002008-12-23T01:55:02.553-08:00Boolean Algebra<b><a name="Boolean Algebra"></a></b><br /> <br />Boolean algebra is one of the most interesting and important algebraic structure which has significant applications in switching<br /> circuits, logic and many branches of computer science and engineering.<br /> <br /> Boolean algebra can be viewed as one of the special type of lattice.<br /> <i> <br />A complemented distributive lattice with 0 and 1 is called <b>Boolean algebra.<br /> </b></i>Generally Boolean algebra is denoted by (B, *, <span style="font-family:Symbol;">Å</span> , ', <b>0</b>, <b>1</b>).<br /> <br /> <b>Example 1 : </b><br /> <br /> ( <i>P</i>(A), <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> , ', <span style="font-family:Symbol;">f, A</span>) is a Boolean algebra. This is an important example of Boolean algebra [In fact the basic properties of<br /> the (P (A), <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> , ' ) led to define the abstract concept of Boolean algebra]. Further, it can be proved that every finite Boolean<br /> algebra must be isomorphic to (P (A), <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> , ' , <span style="font-family:Symbol;">f ,</span> A) for a suitably chosen finite set A. [refer [2]].<br /> <br /> <b>Example 2:</b><br /> <br /> The structure ( B<sup>n</sup> = {0,1}<sup>n</sup> , *, <span style="font-family:Symbol;">Å</span> , <b>1</b>, <b>0</b> ) is a Boolean algebra, where B<sup>n</sup> is an n-fold Cartesian product of<br />{0,1} and the operations *, <span style="font-family:Symbol;">Å</span> , are defined below.<br /> We have B<sup>n</sup> = {( l<sub>1</sub>, l<sub> 2</sub>, … , l<sub> n</sub>) / l<sub> r</sub> = 0 or 1, 1 <span style="font-family:Symbol;">£</span> r <span style="font-family:Symbol;">£</span> n}<br /> (i<sub> 1</sub>, i<sub> 2</sub>, … , i<sub>n</sub>) * (j<sub>1 </sub>, j<sub>2</sub>, … , j<sub>n</sub>) = (min (i<sub> 1</sub>, j<sub>1 </sub>), min (i<sub> 2</sub>, j<sub>2</sub>), .., min (i <sub>n</sub>, j<sub>n</sub>) )<br /> (i<sub>1</sub>, i<sub>2</sub>, … , i<sub>n</sub>) <span style="font-family:Symbol;">Å</span> (j<sub>1</sub>, j<sub>2</sub>,…, j<sub>n</sub>) = (max ( i<sub>1</sub>, j<sub>1</sub>), max ( i<sub>2</sub>, j<sub>2</sub>), … , max ( i<sub> n</sub>, j<sub>n</sub>) )<br /> (l<sub> 1</sub>, l<sub> 2</sub>, l<sub> 3</sub>, … , l<sub> n</sub>)' = (l<sub> 1</sub>', l<sub> 2</sub>', …. , l<sub> n</sub>'),<br /> <br /><br /> <br /><b>1 </b>= (1,1, …, 1) is the greatest element of B<sup>n</sup>.<br /> <b>0</b> = (0, 0,0, … ,0) is the least element of B<sup>n</sup>.<br /> Since B is distributive, B<sup>n</sup> is distributive. From the definition of unary operation " <b>'</b> ",<br /> <br />it is clear that B<sup>n</sup> is complemented. Further, it has <b>0</b> and <b>1</b>. Thus, (B<sup>n</sup>, *, <span style="font-family:Symbol;">Å</span> , ', <b>0</b>,<b>1</b>) is a Boolean algebra.<br /> For the case n = 3 we have,<br /> B<sup>3</sup> = {000, 100, 010, 001, 110, 101, 011, 111}.<br /> The structure of the B<sup>3</sup> is given in the following Hasse diagram.<br /> <br /><b> <br /> </b> <span style="font-size:85%;"> </span> <br />The Boolean algebra (B<sup>n</sup>, *, <span style="font-family:Symbol;">Å</span> , ', 0,1) plays an important role in the construction of switching circuits, electronic circuits and<br /> other applications. Also it can be proved that every finite Boolean algebra is isomorphic to the above Boolean algebra<br /> (B<sup>n</sup>, *, <span style="font-family:Symbol;">Å</span> , ', <b>0</b>,<b>1</b>), for some n. Thus, it is interesting to observe that number of elements in any finite Boolean algebra must be<br /> always 2<sup>n</sup>, for some n.<br /> <br /> <i>Let (B, *, <span style="font-family:Symbol;">Å</span> , ' , 0,1) be a Boolean algebra and S <span style="font-family:Symbol;">Í</span> B. If S contains the elements 0 and 1 and is closed under the<br /> operation *, <span style="font-family:Symbol;">Å</span> and ' then (S, *, <span style="font-family:Symbol;">Å</span> , ', 0,1) is called <b>sub Boolean algebra</b>.<br /> <br /> </i><b> <br />Example 1:</b><br /><br />Consider the Boolean algebra (P ({1,2,3}), <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> , ' ,<span style="font-family:Symbol;">f</span> , {1,2,3})<br /> <br /> <br /><br />Then (S = {<span style="font-family:Symbol;">f</span> , {1}, {2,3}, {1,2,3}}, <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> , ', <span style="font-family:Symbol;">f</span> ,<b> </b> {1,2,3}) is also sub Boolean algebra.<br />Similarly, S = ({<span style="font-family:Symbol;">f</span> ,{3},{1,2},{1,2,3}}, <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> , ', <span style="font-family:Symbol;">f</span> , {1,2,3}) is also sub Boolean algebra.<br /> But (S = ({<span style="font-family:Symbol;">f</span>, {1}, {2,3}, {1,2,3}}, <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È</span> ,' , <span style="font-family:Symbol;">f</span> , {1,2,3})) is not a sub Boolean algebra.<br /> <br />[Find why it is not a sub Boolean algebra].<br /> <br /><br /> <i>If we have two Boolean algebras (B<sub>1</sub>, *<sub>1</sub>, <span style="font-family:Symbol;">Å</span> <sub>1</sub>, ', <b>0<sub>1</sub>, 1<sub>1</sub></b>) and (B<sub>2</sub>, *<sub>2</sub>, <span style="font-family:Symbol;">Å</span> <sub>2</sub>, ''<b>, 0<sub>2</sub>, 1<sub>2</sub></b>) then we can get new Boolean algebra<br /> by taking direct product of these two Boolean algebras. The <b>direct product </b>of these Boolean algebra is the Boolean algebra<br /> (B<sub>1</sub> <span style="font-family:Symbol;">´</span> B<sub>2</sub>, *<sub>3</sub>, <span style="font-family:Symbol;">Å</span> <sub>3</sub>, ''', 0<sub>3</sub>, 1<sub>3</sub>); where for any two elements (a<sub>1</sub>, b<sub>1</sub>) and (a<sub>2</sub>, b<sub>2</sub>) <span style="font-family:Symbol;">Î</span> B<sub>1</sub> <span style="font-family:Symbol;">´</span> B<sub>2</sub>,<br /> </i> <br />(a<sub>1</sub>,b<sub>1</sub>) * (a<sub>2</sub>,b<sub>2</sub>) = (a<sub>1 </sub>*<sub>1 </sub>a<sub>2</sub>, b<sub>1 </sub>*<sub>2 </sub>b<sub>2</sub>)<br /><br />(a<sub>1</sub>,b<sub>1</sub>)<span style="font-family:Symbol;">Å</span> (a<sub>2</sub>,b<sub>2</sub>) = (a<sub>1 </sub><span style="font-family:Symbol;">Å</span> <sub> 1 </sub>a<sub>2 </sub>, b<sub>1 </sub><span style="font-family:Symbol;">Å</span> <sub> 2 </sub>b<sub>2</sub>)<br /> <br />(a<sub>1</sub>,b<sub>1</sub>) ''' = (a<sub>1</sub>', b<sub>1</sub>'')<br /> <br />0<sub>3</sub>=(0<sub>1</sub>,0<sub>2</sub>) <b> </b>and<b> </b> 1<sub>3</sub>=(1<sub>1</sub>,1<sub>2</sub>).<br /> <i> <br />Let (B, *, <span style="font-family:Symbol;">Å</span> , </i>'<i>, 0<sub>B</sub> , 1<sub>B</sub>) and (A, </i><span style="font-family:Symbol;">Ç <i> </i></span> <i> , </i><span style="font-family:Symbol;">È</span><i>, - , 0<sub>A </sub>, 1<sub>A</sub>) be two Boolean algebra. A mapping<br /> <br />f : B<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Image49.gif" width="20" height="14" /> A is called a <b>Boolean homomorphism</b> if f preserves all the Boolean operations, that is,<br /> <br /> </i>f (a * b) = f (a) <span style="font-family:Symbol;">Ç</span> f (b).<br /> f (a <span style="font-family:Symbol;">Å</span> b) = f (a) <span style="font-family:Symbol;">È</span> f (b).<br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Sectio2.gif" width="75" border="0" height="20" /><br /> f (0<sub>B</sub>) = 0<sub>A.</sub><br /> f (1<sub>B</sub>) = 1<sub>A.</sub><br /> <br /> <i> <br />A bijective Boolean homomorphism is called <b>Boolean isomorphism.<br /> <br /> </b></i><br /><b>Exercise:</b><br /><ol><li> Prove that mapping f defined in the exercise 1 of the Section 3.3.3 is a Boolean isomorphism.<br /> </li><li> In every Boolean algebra, prove that (x * y)' = x' <span style="font-family:Symbol;">Å</span> y' and (x <span style="font-family:Symbol;">Å</span> y)' = x' * y', for all x, y.<br /> </li><li> In a Boolean algebra B, for all x, y <span style="font-family:Symbol;">Î</span> B, prove that<br /> x <span style="font-family:Symbol;">£</span><sub> </sub>y <sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Image22.gif" width="22" height="16" /></sub>x * y = 0<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Image22.gif" width="22" height="16" /></sub>x' <span style="font-family:Symbol;">Å</span> y = 1<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Image50.gif" width="22" height="16" /></sub>x * y = x<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Image50.gif" width="22" height="16" /></sub>x <span style="font-family:Symbol;">Å</span> y = y.</li></ol><br /><ol><li><br /></li></ol> <br /> fig ( i ) fig ( ii ) fig ( iii )<br /><ol start="4"><li> Show that in a Boolean algebra the following are equivalent for any a and b.<br /> ( i ) a' <span style="font-family:Symbol;">Ú</span> b = 1<br /> ( ii ) a <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.4/Image/Image23.gif" width="14" height="13" />b' = 0.<br /> </li><li> The Boolean algebras (P (A<sub>3</sub>), <span style="font-family:Symbol;">Ç</span> , <span style="font-family:Symbol;">È,</span> ', A<sub>3</sub>, <span style="font-family:Symbol;">f</span>) and D<sub>30</sub>, where A<sub>3</sub> = {1,2,3} are (Boolean) isomorphic? Justify your<br /> answer.</li></ol>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com2tag:blogger.com,1999:blog-7399621945608861143.post-62650995696040546862008-08-12T04:32:00.000-07:002008-12-23T01:55:02.553-08:00Lattices<div style="text-align: justify;"><meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///D:%5CDOCUME%7E1%5Cshesu04%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C06%5Cclip_filelist.xml"><link rel="Edit-Time-Data" href="file:///D:%5CDOCUME%7E1%5Cshesu04%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C06%5Cclip_editdata.mso"><!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--><title>© Moreniche</title><!--[if gte mso 9]><xml> <o:documentproperties> <o:author>Marcus Polo</o:Author> <o:version>11.9999</o:Version> </o:DocumentProperties> </xml><![endif]--><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> <w:usefelayout/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:SimSun; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:ËÎÌå; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:"\@SimSun"; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:"\@Arial Unicode MS"; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:B; panose-1:0 0 0 0 0 0 0 0 0 0; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:3 0 0 0 1 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:SimSun;} a:link, span.MsoHyperlink {color:blue; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {color:blue; text-decoration:underline; text-underline:single;} p {mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:SimSun;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:52238659; mso-list-template-ids:-1735852288;} @list l1 {mso-list-id:60758012; mso-list-template-ids:-880241786;} @list l2 {mso-list-id:133066294; mso-list-template-ids:-1481367130;} @list l2:level1 {mso-level-start-at:7; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l3 {mso-list-id:332487474; mso-list-template-ids:-984601784;} @list l3:level1 {mso-level-start-at:3; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l4 {mso-list-id:758528941; mso-list-template-ids:719491126;} @list l4:level1 {mso-level-number-format:roman-lower; mso-level-tab-stop:.5in; mso-level-number-position:right; text-indent:-.25in;} @list l5 {mso-list-id:815338869; mso-list-template-ids:-593695876;} @list l6 {mso-list-id:947734086; mso-list-template-ids:947970136;} @list l6:level1 {mso-level-start-at:3; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l7 {mso-list-id:1122184730; mso-list-template-ids:-904128324;} @list l8 {mso-list-id:1190488626; mso-list-template-ids:-1942973984;} @list l9 {mso-list-id:1530414390; mso-list-template-ids:-102326086;} @list l9:level1 {mso-level-start-at:7; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l10 {mso-list-id:1684937800; mso-list-template-ids:2084182182;} @list l10:level1 {mso-level-number-format:roman-lower; mso-level-tab-stop:.5in; mso-level-number-position:right; text-indent:-.25in;} @list l11 {mso-list-id:1814641510; mso-list-template-ids:-774620738;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> </div><p style="text-align: justify;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" spt="75" preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:formulas> <v:path extrusionok="f" gradientshapeok="t" connecttype="rect"> <o:lock ext="edit" aspectratio="t"> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" alt="" style="'width:575.25pt;"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.1\Image\Sectio13.gif"> </v:shape><![endif]--><!--[if !vml]-->
<br /><!--[endif]--><o:p></o:p></p><div style="text-align: justify;"> </div><p style="text-align: justify;" class="MsoNormal"><b><nobr>Lattices</nobr></b>
<br />
<br />In this section we introduce lattices as special type of partial ordered set and we discuss basic properties of lattices and some
<br />important type of special lattices.
<br />
<br /><a name="6.3.1._Lattice_Ordered_Sets"><b>6.3.1. Lattice Ordered Sets</b></a><b>
<br /></b>
<br />In this section we define lattice ordered sets and see some examples.
<br />
<br />
<br /><i>A poset (L, </i><i><span style="font-family: Symbol;">£</span></i> <i>) is called <b>lattice ordered set</b> if for every pair of elements x, y </i><i><span style="font-family: Symbol;">Î</span></i> <i>L, the</i> <i>sup (x, y) and inf (x, y) exist in L.
<br />
<br /></i><b>
<br />Example 1:
<br />
<br /></b>Let S be a nonempty set. Then (<i>P</i>(S), <span style="font-family: Symbol;">Í</span> ) is a lattice ordered set. For (<i>P</i> (S), <span style="font-family: Symbol;">Í</span> ) is a poset. Further for any subsets A, B of S,
<br />inf (A, B) = A <span style="font-family: Symbol;">Ç</span> B <span style="font-family: Symbol;">Î</span> <i>P</i>(S) and sup (A, B) = A <span style="font-family: Symbol;">È </span>B <span style="font-family: Symbol;">Î</span> P(S).
<br />
<br /><b>Example 2:</b>
<br />
<br />Every totally ordered set is a lattice ordered set (Prove !).
<br />
<br /><b>Example 3:
<br />
<br /></b>Consider the set of all positive integer Z<sup>+</sup> with divisor as a relation, i.e., a <span style="font-family: Symbol;">£</span> b if and only if a<span style="font-family: Symbol;">½</span>b.
<br />Then (Z<sup>+ </sup>, <span style="font-family: Symbol;">½</span>) is a poset.
<br />For, if a, b<!--[if gte vml 1]><v:shape id="_x0000_i1026" type="#_x0000_t75" alt="" style="'width:9pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image1.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1026" width="12" height="12" /><!--[endif]--> Z<sup>+</sup>, then inf (a, b) = GCD(a, b) <!--[if gte vml 1]><v:shape id="_x0000_i1027" type="#_x0000_t75" alt="" style="'width:9pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image1.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1027" width="12" height="12" /><!--[endif]-->Z<sup>+ </sup>and sup (a, b) = LCM(a, b) <!--[if gte vml 1]><v:shape id="_x0000_i1028" type="#_x0000_t75" alt="" style="'width:9pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image1.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1028" width="12" height="12" /><!--[endif]-->Z<sup>+</sup>.
<br />Thus, inf (a, b) and sup (a,b) exist in Z<sup>+</sup> for any two element a, b <span style="font-family: Symbol;">Î</span> Z<sup>+</sup>.
<br />Hence (Z<sup>+ </sup>, <span style="font-family: Symbol;">½</span>) is a lattice ordered set. In fact (D<sub>n</sub><sub><span style="font-size: 13.5pt;"> </span></sub>, <span style="font-family: Symbol;">½</span>) ( D<sub>n</sub> denotes the set of all positive divisors of positive number n ) is also
<br />a lattice ordered set.
<br />
<br /><b>
<br />Example 4:
<br /></b>Consider the set B, where B<sup>n </sup>= {(l<sub>1</sub>, l<sub>2</sub>, … , l<sub>n</sub>) / l<sub>i</sub> = 0 or 1, for 1 <span style="font-family: Symbol;">£ </span>r <span style="font-family: Symbol;">£ </span>n}.
<br />Define the relation <span style="font-family: Symbol;">£ </span>' by (i<sub>1</sub>, i<sub>2</sub>, … , i<sub>n</sub>) <span style="font-family: Symbol;">£ </span>' (j<sub>1</sub>, j<sub>2</sub>, … , j<sub>n</sub>) if and only if i<sub>r</sub> <span style="font-family: Symbol;">£ </span>j<sub>r</sub> , 1 <span style="font-family: Symbol;">£ </span>r <span style="font-family: Symbol;">£ </span>n.
<br />Note that here in the expression i<sub>r</sub> <span style="font-family: Symbol;">£</span> j<sub>r</sub>, <span style="font-family: Symbol;">£</span> is usual less than or equal to.
<br />We have already seen in Example 7 of Section 6.2.1 that (B<sub>n</sub>, <span style="font-family: Symbol;">£ </span>') is a poset.
<br />
<br />Observe that
<br />inf [ (i<sub>1</sub>, i<sub>2</sub>, ….. ,i<sub>n</sub>), (j<sub>1</sub>, j<sub>2</sub>, … , j<sub>n</sub>)] = (min (i<sub>1</sub>, j<sub>1</sub>), min (i<sub>2</sub>,j<sub>2</sub>), …. , min (i<sub><span style="font-family: B;">n</span></sub>, j<sub><span style="font-family: B;">n</span></sub>) ) and
<br />sup [ (i<sub>1</sub>, i<sub>2</sub>, … , i<sub><span style="font-family: B;">n</span></sub>), (j<sub>1</sub>, j<sub>2</sub>, … , j<sub><span style="font-family: B;">n</span></sub>)] = (max (i<sub>1</sub>, j<sub>1</sub>), max (i<sub>2</sub>,j<sub>2</sub>), …. , max (i<sub><span style="font-family: B;">n</span></sub>, j<sub><span style="font-family: B;">n</span></sub>) )
<br />
<br />Since min (i<sub>r</sub>, j<sub>r</sub>) and max (i<sub>r</sub>, j<sub>r</sub>) is either 0 or 1,
<br />so, inf { (i<sub>1</sub>, i<sub>2</sub>,… , i<sub>n</sub>), (j<sub>1</sub>,j<sub>2</sub>, .. ,j<sub>n</sub>) } and sup { (i<sub>1</sub>, i<sub>2</sub>, … , i<sub><span style="font-family: B;">n</span></sub>), (j<sub>1</sub>, j<sub>2</sub>, … , j<sub><span style="font-family: B;">n</span></sub>) } exist in B.
<br />Thus, (B<sub>n</sub>, <span style="font-family: Symbol;">£</span> ) is a lattice ordered set.
<br />
<br /><b>Example 5:</b>
<br />Poset represented by the Hasse diagram is not a lattice ordered set since inf (a, b) does not exist.
<br /> <sub><!--[if gte vml 1]><v:shape id="_x0000_i1029" type="#_x0000_t75" alt="" style="'width:183.75pt;height:204.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image003.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.2\Image\Sectio7.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>
<br /><b>
<br />
<br />
<br />Example 6:</b>
<br />Poset represented by the Hasse diagram is not a lattice ordered set as sup (f, g) does not exist.
<br />
<br /> <!--[if gte vml 1]><v:shape id="_x0000_i1030" type="#_x0000_t75" alt="" style="'width:129.75pt;height:186.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image004.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio11.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br /><b>
<br />Exercise </b>:<o:p></o:p></p><div style="text-align: justify;"> </div><ol style="text-align: justify;" start="1" type="1"><li class="MsoNormal">Prove that if (L, <span style="font-family: Symbol;">£</span> ) and (M, <span style="font-family: Symbol;">£ </span>' ) are lattice ordered sets. Then (L <span style="font-family: Symbol;">´</span> M, R) is a lattice ordered set, where (a, b) R (x, y)
<br /> if and only if a <span style="font-family: Symbol;">£</span> x in L and b <span style="font-family: Symbol;">£ </span>' y in M.<o:p></o:p></li><li class="MsoNormal">Check whether the poset represented by the following Hasse diagram that is lattice ordered set or not?<o:p></o:p></li></ol><div style="text-align: justify;"> </div><p style="line-height: 150%; text-align: justify;">
<br /> <!--[if gte vml 1]><v:shape id="_x0000_i1031" type="#_x0000_t75" alt="" style="'width:131.25pt;height:187.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image005.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br />
<br /><b>Remark 1:
<br /></b>Let (L, <span style="font-family: Symbol;">£</span> ) be a lattice ordered set and let x, y <span style="font-family: Symbol;">Î</span> L. Then the following are equivalent.
<br />(i) x <span style="font-family: Symbol;">£</span> y
<br />(ii) sup (x, y) = y
<br />(iii) inf (x, y) = x
<br /><b>
<br />Proof</b>:
<br />
<br />( i )<sub><!--[if gte vml 1]><v:shape id="_x0000_i1032" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1032" width="20" height="16" /><!--[endif]--> </sub>( ii )
<br />Let x <span style="font-family: Symbol;">£</span> y …… (1)
<br />We have, y <span style="font-family: Symbol;">£</span> y , for all y <span style="font-family: Symbol;">Î</span> L. …… (2)
<br />From (1) and (2), we have y is an upper bound of (x, y).
<br />Therefore, sup (x, y ) <span style="font-family: Symbol;">£</span> y (by definition of superimum).
<br />But, y <span style="font-family: Symbol;">£</span> sup (x, y).
<br />Therefore, y = sup (x, y) (since <span style="font-family: Symbol;">£</span> is anti - symmetric).
<br />( ii ) <sub><!--[if gte vml 1]><v:shape id="_x0000_i1033" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1033" width="20" height="16" /><!--[endif]--></sub>( iii )
<br />Given that sup (x, y) = y. Therefore we have x <span style="font-family: Symbol;">£ </span>y.
<br />Also, we have x <span style="font-family: Symbol;">£</span> x.
<br />Therefore, x is a lower bound for x and y.
<br />Thus, x <span style="font-family: Symbol;">£</span> inf (x, y).
<br />But, we have inf (x, y) <span style="font-family: Symbol;">£</span> x.
<br />Hence, x = inf (x, y) ( since <span style="font-family: Symbol;">£</span> is anti-symmetric).
<br />( iii ) <sub><!--[if gte vml 1]><v:shape id="_x0000_i1034" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1034" width="20" height="16" /><!--[endif]--></sub>( i )
<br />
<br />Given that x = inf (x, y).
<br />Therefore, by the definition of infimum, x <span style="font-family: Symbol;">£</span> y.
<br /> <!--[if !supportLineBreakNewLine]--><b>
<br /><nobr><a name="6.3.2_Algebraic_Lattice">6.3.2 Algebraic Lattice</a>
<br /></nobr></b>
<br /><nobr>In this section we define algebraic lattice by using two binary operations * (meet) and <span style="font-family: Symbol;">Å</span> (join). Then we shall prove that lattice
<br />ordered sets and algebraic lattices are equivalent.
<br />
<br /><i>
<br />An <b>algebraic lattice</b> (L, *, </i><i><span style="font-family: Symbol;">Å </span>) is a non empty set L with two binary operations * (meet) and </i><span style="font-family: Symbol;">Å</span><i> (join), which satisfy the
<br />following conditions for all x, y, z <!--[if gte vml 1]><v:shape id="_x0000_i1036" type="#_x0000_t75" alt="" style="'width:9pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image1.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1036" width="12" border="0" height="12" /><!--[endif]-->L.
<br /></i>
<br /><span style="" lang="FR">L1. x * y = y * x, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y = y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> x (Commutative)
<br />
<br />L2. x * (y*z) = (x * y) * z, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z) = (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z (Associative)
<br />
<br />L3. x * (x </span><span style="font-family: Symbol;">Å </span><span style="" lang="FR">y) = x, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (x * y) = x (Absorption)
<br />
<br />L4. x * x = x, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> x = x (Idempotent)
<br />
<br />
<br /><b>Theorem 3.2.1:
<br /></b>Let (L, </span><span style="font-family: Symbol;">£</span><span style="" lang="FR">) be a lattice ordered set. </span>If we define x * y = inf (x, y), x <span style="font-family: Symbol;">Å</span> y = sup (x, y) then (L, *, <span style="font-family: Symbol;">Å </span>) is an algebraic lattice.
<br />
<br /><b>Proof:
<br /></b>Given that (L, <span style="font-family: Symbol;">£</span> ) is a lattice ordered set and x * y = inf (x, y) and x <span style="font-family: Symbol;">Å</span> y = sup (x, y).
<br />Now we shall prove that * and <span style="font-family: Symbol;">Å</span> satisfy the commutative, associative, absorption and idempotent laws.
<br /><b>
<br /></b><b><span style="" lang="FR">Commutative
<br /></span></b><span style="" lang="FR">
<br />x * y = inf (x, y) = inf (y, x) = y * x.
<br />x </span><span style="font-family: Symbol;">Å </span><span style="" lang="FR">y = sup (x, y) = sup (y, x) = y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> x.
<br /><b>
<br />Associative
<br /></b>
<br />x * (y * z) = inf (x, (y * z))
<br /> = inf (x, inf (y,z))
<br /> = inf (x,y,z)
<br /> = inf ( inf (x, y), z))
<br /> = inf ((x*y), z)
<br /> = (x*y) *z.
<br />
<br />Now, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z) = sup (x, (y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z))
<br /> = sup (x, sup (y,z))
<br /> = sup (x, y,z)
<br /> = sup (sup (x,y), z)
<br /> = sup ((x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y), z)
<br /> = (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z.
<br /><b>
<br />Absorption</b>
<br />
<br />Now, x * (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) = inf (x, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y)
<br /> = inf (x, sup (x, y))
<br /> = x [since x </span><span style="font-family: Symbol;">£</span><span style="" lang="FR"> sup (x, y)]
<br />and
<br />x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (x * y) = sup (x, x * y)
<br /> = sup (x, inf (x, y))
<br /> = x [since inf (x, y) </span><span style="font-family: Symbol;">£</span><span style="" lang="FR"> x ].
<br /><b>
<br /></b></span><b>Idempotent
<br /></b>
<br />We have, x * x = inf (x, x) = x and x <span style="font-family: Symbol;">Å</span> x = sup (x, x) = x.
<br />Hence (L, * , <span style="font-family: Symbol;">Å </span>) is an algebraic lattice.
<br />
<br />
<br />
<br /><b>Theorem 3.2.2:
<br /></b>Let (L, *, <span style="font-family: Symbol;">Å</span> ) be an algebraic lattice. If we define x <span style="font-family: Symbol;">£</span> y <sub><!--[if gte vml 1]><v:shape id="_x0000_i1037" type="#_x0000_t75" alt="" style="'width:16.5pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image007.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image22.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image007.gif" shapes="_x0000_i1037" width="22" border="0" height="16" /><!--[endif]--></sub>x * y = x or x <span style="font-family: Symbol;">£</span> y <sub><!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75" alt="" style="'width:16.5pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image007.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image22.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image007.gif" shapes="_x0000_i1038" width="22" border="0" height="16" /><!--[endif]--></sub>x <span style="font-family: Symbol;">Å</span> y = y, then (L, <span style="font-family: Symbol;">£</span> ) is a lattice ordered set.
<br />
<br /><b>Proof:
<br /></b>Given that (L, *, <span style="font-family: Symbol;">Å</span> ) is an algebraic lattice and x <span style="font-family: Symbol;">£</span> y <sub><!--[if gte vml 1]><v:shape id="_x0000_i1039" type="#_x0000_t75" alt="" style="'width:16.5pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image007.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image22.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image007.gif" shapes="_x0000_i1039" width="22" border="0" height="16" /><!--[endif]--></sub>x * y = x or x <span style="font-family: Symbol;">Å</span> y = y.
<br />We shall now prove that (L, <span style="font-family: Symbol;">£</span> ) is a poset and inf (x, y) and sup (x, y) exist in L, for all x, y in L.
<br />
<br />
<br /><b><span style="font-family: Symbol;">£</span> is reflexive
<br /></b>Since x * x = x, for all x <!--[if gte vml 1]><v:shape id="_x0000_i1040" type="#_x0000_t75" alt="" style="'width:9pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image1.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1040" width="12" border="0" height="12" /><!--[endif]-->L (by indempotent of *).
<br />We have by definition of <span style="font-family: Symbol;">£</span> , x <span style="font-family: Symbol;">£</span> x, for all x <span style="font-family: Symbol;">Î</span> L.
<br />Therefore, <span style="font-family: Symbol;">£</span> is reflexive.
<br />
<br /><b>
<br /></b><b><span style="font-family: Symbol;">£</span> is anti-symmetric</b>
<br />If x <span style="font-family: Symbol;">£</span> y and y <span style="font-family: Symbol;">£</span> x in L, then by definition of <span style="font-family: Symbol;">£</span> , we have x * y = x and y * x = y.
<br />But * satisfies commutative law, so, we have x * y = y * x.
<br />Therefore, x = y.
<br />Hence, <span style="font-family: Symbol;">£</span> is anti-symmetric.
<br />
<br /><b>
<br /></b><b><span style="font-family: Symbol;">£</span> is transitive</b>
<br />If x <span style="font-family: Symbol;">£</span> y and y <span style="font-family: Symbol;">£</span> z, then by the definition of <span style="font-family: Symbol;">£</span> , we have x * y = x and y * z = y.
<br />Therefore, x * z = (x * y) * z
<br /> = x * (y * z) [by associativity of *]
<br /> = x * y [by definition <span style="font-family: Symbol;">£</span> )]
<br /> = x [by definition of <span style="font-family: Symbol;">£</span> ]
<br />Hence, by definition of <span style="font-family: Symbol;">£</span> , we have x <span style="font-family: Symbol;">£</span> z.
<br />Thus, <span style="font-family: Symbol;">£</span> is transitive.
<br /><b>
<br />sup (x, y) and inf (x, y) exist in L</b>
<br />We shall now show that inf (x, y) = x * y and sup (x, y) = x <span style="font-family: Symbol;">Å</span> y.
<br />Now by absorption law, we have
<br />x = x * (x <span style="font-family: Symbol;">Å </span>y) and y = y * (x <span style="font-family: Symbol;">Å</span> y)
<br />Then by the definition of <span style="font-family: Symbol;">£</span> , we have x <span style="font-family: Symbol;">£</span> x <span style="font-family: Symbol;">Å</span> y and y <span style="font-family: Symbol;">£</span> x <span style="font-family: Symbol;">Å</span> y
<br />Therefore, x <span style="font-family: Symbol;">Å</span> y is an upper bound for x and y.
<br />Let z be any upper bound for x and y in L.
<br />Then x <span style="font-family: Symbol;">£</span> z and y <span style="font-family: Symbol;">£</span> z. So, by definition of <span style="font-family: Symbol;">£</span> ,
<br />we have x <span style="font-family: Symbol;">Å</span> z = z and y <span style="font-family: Symbol;">Å</span> z = z …… ( 1 )
<br />Therefore, (x <span style="font-family: Symbol;">Å</span> y) <span style="font-family: Symbol;">Å</span> z = x <span style="font-family: Symbol;">Å</span> (y <span style="font-family: Symbol;">Å</span> z) [by associative law]
<br /> = x <span style="font-family: Symbol;">Å </span>z [by ( 1 )]
<br /> = z [by ( 1 )].
<br />Thus, by definition of <span style="font-family: Symbol;">£</span> , we have x <span style="font-family: Symbol;">Å</span> y <span style="font-family: Symbol;">£</span> z. that is, x <span style="font-family: Symbol;">Å</span> y is related to every upper bound of x and y. Hence sup (x, y) = x <span style="font-family: Symbol;">Å</span> y.
<br />Similarly, we can show that inf (x, y) = x * y.
<br />Thus, x * y and x <span style="font-family: Symbol;">Å</span> y exists for every x, y <span style="font-family: Symbol;">Î</span> L.
<br />Hence, we have (L, <span style="font-family: Symbol;">£</span> ) is lattice ordered set.
<br />
<br />
<br />
<br />
<br /><b>Remark 1:</b>
<br />
<br />From the Theorem 3.2.1 and the Theorem 3.2.2, we see that if we have lattice ordered set (L, <span style="font-family: Symbol;">£</span> ) then we can get an algebraic
<br />lattice (L, *, <span style="font-family: Symbol;">Å</span> ) and conversely. Hence, we conclude that the algebraic lattice and a lattice ordered sets are equivalent system.
<br />Thus, here after we shall say simply lattice to mean both.
<br />
<br />In an algebraic system it is better convenience for imposing further conditions on the binary operations. Hence developing
<br />structural concepts will be much easier than the ordered system. In fact, it is one of the motivation to view a lattice ordered set as
<br />an algebraic lattice.
<br />
<br />
<br /> <!--[if !supportLineBreakNewLine]--></nobr><b><!--[endif]--></b><o:p></o:p></p><div style="text-align: justify;"> </div><p style="line-height: 150%; text-align: justify;"><a name="6.3.3_Sublattice,_Direct_Product_and_Hom"><b><nobr>6.3.3 Sublattice, Direct Product and Homomorphism</nobr></b></a><nobr>
<br />In this section we discuss sublattices of a lattice, direct product of two lattices and homomorphism between two lattices.
<br />
<br /><a name="6.3.3.1_Sublattices"><b>6.3.3.1 Sublattices</b></a><b>
<br /></b>
<br />Substructure helps to know more about the whole structure. So, here we discuss about sublattice of a lattice.
<br /><i>Let (L, *, </i><i><span style="font-family: Symbol;">Å</span> ) be a lattice and let S be subset of L. The substructure (S, *, </i><i><span style="font-family: Symbol;">Å</span>) is a <b>sublattice</b> of (L, *, </i><i><span style="font-family: Symbol;">Å</span> ) if and only if S is
<br />closed under both operations * and </i><span style="font-family: Symbol;">Å</span><i>.
<br /></i>
<br /><b>Remark 1:
<br /></b> A subset S in a lattice (L, *, <span style="font-family: Symbol;">Å</span> ) is said to be sublattice, for a, b <!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75" alt="" style="'width:9pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image1.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1042" width="12" border="0" height="12" /><!--[endif]-->S, if a * b = c in L and a <span style="font-family: Symbol;">Å</span> b = d in L, then c, d must
<br />necessary exist in S also.
<br />
<br /><b>Example 1:
<br /></b>Consider the lattice L represented by the following Hasse diagram.
<br />
<br />
<br />
<br />Here the substructure S<sub>1</sub> represented by the Hasse diagram given below is not a sublattice, for inf (a, b) = 0 in L, which does not
<br />belong to S<sub>1</sub>.
<br /> <!--[if gte vml 1]><v:shape id="_x0000_i1044" type="#_x0000_t75" alt="" style="'width:217.5pt;height:117pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image009.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio14.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br /> It is clear that the substructure S<sub>2</sub> represented by the Hasse diagram given below is sublattice of L.
<br /> <!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75" alt="" style="'width:177pt;height:158.25pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image010.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio15.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br />
<br /> It is interesting to note that the substructure <sub>3</sub> of L represented by the in the following Hasse diagram is not a sublattice but it is a
<br />lattice on its own. So, it is a lattice without being a sublattice. <!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75" alt="" style="'width:243pt;height:222pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image011.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio16.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br /><b>
<br />
<br />Exercise:</b><o:p></o:p></nobr>ind all the sub lattices of (D<sub>30</sub>, <span style="font-family: Symbol;">½</span>).<o:p></o:p></p><div style="text-align: justify;"> </div><ol style="text-align: justify;" start="1" type="1"><li class="MsoNormal">Let L be a lattice and let a < style="font-family: Symbol;">Î</span> L / a <span style="font-family: Symbol;">£</span> x <span style="font-family: Symbol;">£</span> b }. Prove
<br /> that [a,b] is a sublattice.</li></ol><p style="line-height: 150%; text-align: justify;"><span style="font-size: 10pt; line-height: 150%;"><!--[endif]-->
<br /></span><b>
<br /><a name="6.3.3.2_Direct_Product_of_Lattices"><nobr>6.3.3.2 Direct Product of Lattices</nobr></a>
<br /></b>
<br /><nobr>
<br />From given two lattices, we can always construct a new lattice by taking Cartesian product of the given two lattices. So, in this
<br />section we discuss about product of two lattices.
<br />
<br />
<br />Let (L, *, <span style="font-family: Symbol;">Å</span> ) and (M,<!--[if gte vml 1]><v:shape id="_x0000_i1048" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1048" width="14" border="0" height="13" /><!--[endif]-->,<!--[if gte vml 1]><v:shape id="_x0000_i1049" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image24.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1049" width="14" border="0" height="13" /><!--[endif]-->) be two lattices. Consider the Cartesian product of L and M, that is,
<br />L <span style="font-family: Symbol;">´</span> M = {(x, y) / x <span style="font-family: Symbol;">Î</span> L, y <span style="font-family: Symbol;">Î</span> M}.
<br />Define operations <span style="font-family: Symbol;">D</span> and <span style="font-family: Symbol;">Ñ</span> in L <span style="font-family: Symbol;">´</span> M, by (x, y) <span style="font-family: Symbol;">D</span> (a, b) = (x * a, y<!--[if gte vml 1]><v:shape id="_x0000_i1050" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1050" width="14" border="0" height="13" /><!--[endif]--> b) and (x, y) <span style="font-family: Symbol;">Ñ</span> (a, b) = (x <span style="font-family: Symbol;">Å</span> a, y<!--[if gte vml 1]><v:shape id="_x0000_i1051" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image24.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1051" width="14" border="0" height="13" /><!--[endif]-->b), then we shall prove
<br />that ( L <span style="font-family: Symbol;">´</span> M, <span style="font-family: Symbol;">D</span> ,<span style="font-family: Symbol;">Ñ</span> ) is a lattice.
<br />
<br />
<br /><b><span style="font-family: Symbol;">D</span></b> <b>and </b><b><span style="font-family: Symbol;">Ñ</span> are commutative.</b>
<br />
<br />By definition (x, y) <span style="font-family: Symbol;">D</span> (a, b) = (x * a, y <!--[if gte vml 1]><v:shape id="_x0000_i1052" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1052" width="14" border="0" height="13" /><!--[endif]-->b)
<br /> = (a * x, b <!--[if gte vml 1]><v:shape id="_x0000_i1053" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1053" width="14" border="0" height="13" /><!--[endif]-->y) (since * and <!--[if gte vml 1]><v:shape id="_x0000_i1054" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1054" width="14" border="0" height="13" /><!--[endif]-->are commutative)
<br /> = (a, b) <span style="font-family: Symbol;">D</span> (x, y) (by definition <span style="font-family: Symbol;">D</span> ).
<br />
<br />Similarly, (x, y) <span style="font-family: Symbol;">Ñ</span> (a, b) = (x <span style="font-family: Symbol;">Å</span> a, y <!--[if gte vml 1]><v:shape id="_x0000_i1055" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image24.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1055" width="14" border="0" height="13" /><!--[endif]-->b)
<br /> = (a <span style="font-family: Symbol;">Å</span> x, b <!--[if gte vml 1]><v:shape id="_x0000_i1056" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image24.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1056" width="14" border="0" height="13" /><!--[endif]-->y)
<br /> = (a, b) <span style="font-family: Symbol;">Ñ</span> (x, y).
<br />
<br />Hence, commutative law holds good for both operations <span style="font-family: Symbol;">D</span> and <span style="font-family: Symbol;">Ñ</span> .
<br />
<br /><b><span style="font-family: Symbol;">D</span></b> <b>and </b><b><span style="font-family: Symbol;">Ñ</span> are associative </b>
<br />
<br />[ ( x, y) <span style="font-family: Symbol;">D</span> (a, b) ) <span style="font-family: Symbol;">D</span> (u, v)] = [ (x * a, y <!--[if gte vml 1]><v:shape id="_x0000_i1057" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1057" width="14" border="0" height="13" /><!--[endif]-->b) ] <span style="font-family: Symbol;">D</span> (u, v) (by definition of <span style="font-family: Symbol;">D</span> )
<br /> = [ (x * a ) * u, (y <!--[if gte vml 1]><v:shape id="_x0000_i1058" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1058" width="14" border="0" height="13" /><!--[endif]-->b) <!--[if gte vml 1]><v:shape id="_x0000_i1059" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1059" width="14" border="0" height="13" /><!--[endif]-->v] (again by definition of <span style="font-family: Symbol;">D</span> )
<br /> = [ x * (a * u), y <!--[if gte vml 1]><v:shape id="_x0000_i1060" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1060" width="14" border="0" height="13" /><!--[endif]-->(b <!--[if gte vml 1]><v:shape id="_x0000_i1061" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1061" width="14" border="0" height="13" /><!--[endif]-->v) ] (since * and <!--[if gte vml 1]><v:shape id="_x0000_i1062" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1062" width="14" border="0" height="13" /><!--[endif]-->are associative)
<br /> = [ (x, y) <span style="font-family: Symbol;">D</span> (a * u, b <!--[if gte vml 1]><v:shape id="_x0000_i1063" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1063" width="14" border="0" height="13" /><!--[endif]-->v) ] (by definition of <span style="font-family: Symbol;">D</span> )
<br /> = [ (x, y) <span style="font-family: Symbol;">D</span> [ (a, b) <span style="font-family: Symbol;">D</span> (u, v) ] (by definition of <span style="font-family: Symbol;">D</span> )
<br />
<br />Similarly we can show that
<br />[ (x, y) <span style="font-family: Symbol;">Ñ</span> (a, b) ] <span style="font-family: Symbol;">Ñ</span> (u, v) = (x, y) <span style="font-family: Symbol;">Ñ</span> [ (a, b) <span style="font-family: Symbol;">Ñ</span> (u, v) ]
<br />Thus, associative law hold good for both operations and <span style="font-family: Symbol;">Ñ</span> in L <span style="font-family: Symbol;">´</span> M.
<br />
<br /><b>
<br />Absorption
<br /></b>
<br />(x, y) <span style="font-family: Symbol;">D</span> [ (x, y) <span style="font-family: Symbol;">Ñ</span> (a, b) ] = (x, y) <span style="font-family: Symbol;">D</span> [ x <span style="font-family: Symbol;">Å</span> a, y <!--[if gte vml 1]><v:shape id="_x0000_i1064" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image24.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1064" width="14" border="0" height="13" /><!--[endif]-->b]
<br /> = [ x * (x <span style="font-family: Symbol;">Å</span> a), y<!--[if gte vml 1]><v:shape id="_x0000_i1065" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1065" width="14" border="0" height="13" /><!--[endif]--> (y <!--[if gte vml 1]><v:shape id="_x0000_i1066" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image24.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1066" width="14" border="0" height="13" /><!--[endif]-->b) ] (by definition of <span style="font-family: Symbol;">D</span> and <span style="font-family: Symbol;">Ñ</span> )
<br /> = (x, y ) (by absorption law in L and M).
<br />Therefore, absorption law hold good in L <span style="font-family: Symbol;">´</span> M.
<br /><b>
<br />Idempotent
<br /></b>
<br />(x, y) <span style="font-family: Symbol;">D</span> (x, y) = [ x * x, y <!--[if gte vml 1]><v:shape id="_x0000_i1067" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1067" width="14" border="0" height="13" /><!--[endif]-->y]
<br /> = (x, y) (since * and <!--[if gte vml 1]><v:shape id="_x0000_i1068" type="#_x0000_t75" alt="" style="'width:10.5pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1068" width="14" border="0" height="13" /><!--[endif]-->satisfy idempotent law)
<br />Similarly, (x, y) <span style="font-family: Symbol;">Ñ</span> (x, y) = (x, y).
<br />
<br />Hence, (L <span style="font-family: Symbol;">´</span> M, <span style="font-family: Symbol;">D</span> , <span style="font-family: Symbol;">Ñ</span> ) is an algebraic lattice.
<br />Thus, (L <span style="font-family: Symbol;">´</span> M, <span style="font-family: Symbol;">D</span> , <span style="font-family: Symbol;">Ñ</span> ) is a lattice.
<br />
<br /><b>Remark 1:
<br /></b>
<br />If (L, *, <span style="font-family: Symbol;">Å</span> ) is a lattice, then L² = L x L is a lattice. In general one can show that
<br />L<sup>n</sup> = L <span style="font-family: Symbol;">´</span> L <span style="font-family: Symbol;">´</span> L <span style="font-family: Symbol;">´</span> …. <span style="font-family: Symbol;">´</span> L (n times) is a lattice.
<br />
<br />In the finite lattices, B<sup>n</sup> is a very important lattices, where B = {0,1}, which has rich structural property and will play very
<br />important role in the applications.
<br />
<br />Let (L, <span style="font-family: Symbol;">£</span> ) and (M, <span style="font-family: Symbol;">£ </span>' ) be two lattices, then we have already seen that (L <span style="font-family: Symbol;">´</span> M, R) where (x<sub>1</sub>, y<sub>1</sub>) R (x<sub>2</sub>, y<sub>2</sub>) if and only if
<br />x<sub>1 </sub><span style="font-family: Symbol;">£</span> x<sub>2</sub> and y<sub>1</sub> <span style="font-family: Symbol;">£ </span>' y<sub>2</sub>, is a poset. It can be proved that
<br />
<br />(L <span style="font-family: Symbol;">´</span> M, R) is a lattice ordered set. [ Prove !]
<br /> <!--[if !supportLineBreakNewLine]--></nobr><b>
<br /><nobr><a name="6.3.3.3_Homomorphism">6.3.3.3 Homomorphism</a>
<br /></nobr></b>
<br /><nobr>
<br />In this section to understand the structural similarity between two lattices we define homomorphism between two lattices and we
<br />discuss about it.
<br />
<br /><i>
<br />Let</i> (L, *, <span style="font-family: Symbol;">Å</span> ) <i>and</i> (M, <!--[if gte vml 1]><v:shape id="_x0000_i1070" type="#_x0000_t75" alt="" style="'width:11.25pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1070" width="15" border="0" height="12" /><!--[endif]-->,<!--[if gte vml 1]><v:shape id="_x0000_i1071" type="#_x0000_t75" alt="" style="'width:11.25pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image26.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1071" width="15" border="0" height="12" /><!--[endif]-->) <i>be two lattices. A function</i> f : L<span style="font-family: Symbol;"> ®</span> M <i>is called a</i> <b><i>Lattice homomorphism</i></b><i> if
<br /></i>f (a * b) = f (a) <!--[if gte vml 1]><v:shape id="_x0000_i1072" type="#_x0000_t75" alt="" style="'width:11.25pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image23.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1072" width="15" border="0" height="12" /><!--[endif]-->f (b) and
<br />f (a <span style="font-family: Symbol;">Å</span> b) = f (a)<!--[if gte vml 1]><v:shape id="_x0000_i1073" type="#_x0000_t75" alt="" style="'width:11.25pt;height:9pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image26.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1073" width="15" border="0" height="12" /><!--[endif]--> f (b), <i>for all</i> a, b <span style="font-family: Symbol;">Î</span> L,
<br /><i>
<br />and it is called <b>order preserving</b> if x </i><i><span style="font-family: Symbol;">£</span><sub> </sub>y in L implies </i>f (x) <i><sub><!--[if gte vml 1]><v:shape id="_x0000_i1074" type="#_x0000_t75" alt="" style="'width:9pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image20.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1074" width="12" border="0" height="16" /><!--[endif]--></sub></i>' f (y)<i>, where </i><span style="font-family: Symbol;">£</span><i><sub> </sub>is an order relation in L and <sub><!--[if gte vml 1]><v:shape id="_x0000_i1075" type="#_x0000_t75" alt="" style="'width:9pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image20.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1075" width="12" border="0" height="16" /><!--[endif]--></sub>' is an order
<br />relation in M.
<br />
<br />A bijecitve homomorphism is called <b>isomorphism</b>.
<br /></i>
<br />
<br /><b>Example 1:</b>
<br />
<br />Consider the lattices D<sub>6</sub> = {1,2,3,6} and D<sub>30</sub> = {1,2,3,5,6,10,15,30}. We can show that there exist a homomorphism <i>f </i>
<br />between D<sub>6</sub> and D<sub>30</sub>.
<br />
<br />Define a mapping f from D<sub>6</sub> into D<sub>30</sub> by
<br /><i>f</i> (1) = 1,
<br /><i>f</i>(2) = 6,
<br /><i>f</i> (3) = 15 and
<br /><i>f</i> (6) = 30.
<br />
<br />Then <i>f</i> (1 * 2) = <i>f</i>(1) = 1.
<br /><i>f</i> (1) * f(2) = 1 * 6 = 1.
<br />[Note that in both D<sub>6</sub> and D<sub>30 </sub> a * b and a <span style="font-family: Symbol;">Å </span>b are GCD and LCM of two element a, b]
<br />Similarly, <i>f</i> (1 * 3) = <i>f</i>(1) = <i>f</i>(1) * <i>f</i>(3).
<br /><i>f</i>(1 * 6) = <i>f</i>(1) * <i>f </i>(6) = <i>f</i>(1).
<br /><i>f</i>(2 * 3) = <i>f</i>(2) * <i>f</i>(3) = <i>f</i>(1).
<br /><i>f</i>(2 * 6) = <i>f</i>(2) * <i>f</i>(6) = <i>f</i>(2).
<br /><i>f</i>(3 * 6) = <i>f</i>(3) * <i>f</i>(6) = <i>f</i>(3).
<br />
<br />
<br />Similarly we can prove that
<br /><i>f</i> (x <span style="font-family: Symbol;">Å </span>y) = <i>f</i> (x) <span style="font-family: Symbol;">Å</span> <i>f</i> (y) for all x, y <!--[if gte vml 1]><v:shape id="_x0000_i1076" type="#_x0000_t75" alt="" style="'width:9.75pt;height:9.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image18.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1076" width="13" border="0" height="13" /><!--[endif]-->D<sub>6</sub> .
<br />
<br />
<br />Thus, <i>f</i> is homomorphism.
<br />
<br />Note that <i>f</i> is not onto but <i>f </i>is one-one.
<br />
<br />Hence f is not an isomorphism.
<br />
<br /><b>
<br />Exercise 1:</b>
<br />
<br />It is very interesting to prove that the mapping f defined from B<sup>n</sup> into <i>P</i>(A<sub>n</sub>), where
<br />
<br />A<sub>n</sub> = {1,2, …. , n} by <i>f</i> (i<sub>1</sub>, i<sub>2</sub>, …. , i<sub>n</sub>) = { k / i<sub>k</sub> <span style="font-family: Symbol;">¹ </span>0} is a homomorphism.
<br /> <!--[if !supportLineBreakNewLine]--></nobr><b><a name="6.3.3.4_Special_Lattices">
<br /><nobr>6.3.3.4 Special Lattices</nobr></a>
<br /></b>
<br /><nobr>In this section we shall discuss some of the special types of lattices which in turn help to define the Boolean algebra.
<br />
<br />
<br /><a name="6.3.3.4.1_Isotone,_Distributive_and_Modu"><b>6.3.3.4.1 Isotone, Distributive and Modular Inequalities</b></a><b>
<br />
<br /></b>In this subsection we shall prove that in every lattice the operation * and <span style="font-family: Symbol;">Å</span> are isotone and distributive and modular inequalities
<br />holds good.
<br /><b>
<br />Lemma 3.4.1.1:
<br /></b>
<br />In every lattice L the operation * and <span style="font-family: Symbol;">Å</span> are isotone,
<br />
<br />i.e., if y <span style="font-family: Symbol;">£</span> z <span style="font-family: Symbol;">Þ</span> x * y <span style="font-family: Symbol;">£</span> x * z and x <span style="font-family: Symbol;">Å</span> y <span style="font-family: Symbol;">£</span> x <span style="font-family: Symbol;">Å</span> z.
<br /><b>
<br />Proof:</b>
<br />
<br />x * y = (x * x) * (y * z) (since x * x = x and since y <span style="font-family: Symbol;">£</span><sub> </sub>z, y * z = y)
<br /> = (x * y) * (x * z) (by associative and commutative of *).
<br /><sub><!--[if gte vml 1]><v:shape id="_x0000_i1078" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1078" width="20" border="0" height="16" /><!--[endif]--></sub><span style="" lang="FR">x * y </span><span style="font-family: Symbol;">£</span><span style="" lang="FR"> x * z (since a </span><span style="font-family: Symbol;">£</span><sub><span style="" lang="FR"> </span></sub><span style="" lang="FR">b </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1079" type="#_x0000_t75" alt="" style="'width:16.5pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image007.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image22.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image007.gif" shapes="_x0000_i1079" width="22" border="0" height="16" /><!--[endif]--></sub><span style="" lang="FR">a * b = a).
<br />
<br />Also, x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z = (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> x) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z)
<br /> = (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z)
<br /></span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1080" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1080" width="20" border="0" height="16" /><!--[endif]--></sub><span style="" lang="FR">x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y </span><span style="font-family: Symbol;">£</span><span style="" lang="FR"> x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z.
<br /></span>Hence the lemma.
<br />
<br /><b>Theorem 3.4.1.2:
<br /></b>The elements of an arbitrary lattice satisfy the following inequalities <o:p></o:p></nobr></p><div style="text-align: justify;"> </div><ol style="text-align: justify;" start="1" type="i"><li class="MsoNormal"><span style="" lang="FR">x * (y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z) </span><span style="font-family: Symbol;">³</span><span style="" lang="FR"> (x * y) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (x * z)
<br /> x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (y * z) </span><span style="font-family: Symbol;">£</span><span style="" lang="FR"> (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) * (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z) (distributive inequalities.)<o:p></o:p></span></li><li class="MsoNormal"><span style="" lang="FR">x </span><span style="font-family: Symbol;">³</span><span style="" lang="FR"> z </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1081" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1081" width="20" border="0" height="16" /><!--[endif]--></sub><span style="" lang="FR">x * (y </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z) </span><span style="font-family: Symbol;">³</span><span style="" lang="FR"> (x * y) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (x * z) = (x * y) </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z
<br /> x </span><span style="font-family: Symbol;">£ </span><span style="" lang="FR">z </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1082" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1082" width="20" border="0" height="16" /><!--[endif]--></sub><span style="" lang="FR">x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> (y * z) </span><span style="font-family: Symbol;">£</span><span style="" lang="FR"> (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) * (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> z) = (x </span><span style="font-family: Symbol;">Å</span><span style="" lang="FR"> y) * x (modular inequalities.)<o:p></o:p></span></li></ol><div style="text-align: justify;"> </div><p style="line-height: 150%; text-align: justify;"><b>
<br />Proof:
<br />
<br />Claim: </b><i>Distributive inequalities are true in a lattice.
<br /></i>
<br />By definition of <span style="font-family: Symbol;">Å, </span>y <span style="font-family: Symbol;">£ </span>y <span style="font-family: Symbol;">Å </span>z and z <span style="font-family: Symbol;">£ </span>y <span style="font-family: Symbol;">Å </span>z.
<br />By isotone property
<br />
<br />we have x * y <span style="font-family: Symbol;">£ </span>x * (y <span style="font-family: Symbol;">Å </span>z) ............... (1)
<br />and x * z <span style="font-family: Symbol;">£ </span>x * (y <span style="font-family: Symbol;">Å </span>z) ............... (2)
<br />
<br />From (1) and (2) we observe that x * (y <span style="font-family: Symbol;">Å</span> z) is an upper bound for x * y and x * z.
<br />
<br />Hence, (x * y) <span style="font-family: Symbol;">Å</span> (x * z) <span style="font-family: Symbol;">£</span> x * (y <span style="font-family: Symbol;">Å</span> z).
<br />By duality principle, we have (x <span style="font-family: Symbol;">Å</span> y) * (x <span style="font-family: Symbol;">Å</span> z) <span style="font-family: Symbol;">³</span> x <span style="font-family: Symbol;">Å</span> (y * z).
<br />
<br />ii) It is given that z <span style="font-family: Symbol;">£</span> x, then by Theorem 3.2.2 z * x = z.
<br />From the inequality ( i ) we have x* (y <span style="font-family: Symbol;">Å </span>z) <span style="font-family: Symbol;">³</span> (x*y)<span style="font-family: Symbol;"> Å (</span>x<span style="font-family: Symbol;">*</span>z<span style="font-family: Symbol;">).
<br /></span>Therefore, x * (y <span style="font-family: Symbol;">Å</span> z) <span style="font-family: Symbol;">³</span> (x * y) <span style="font-family: Symbol;">Å</span> z.
<br />Since x <span style="font-family: Symbol;">£</span> z, we have x <span style="font-family: Symbol;">Å</span> z = z.
<br />Thus, from the inequality x <span style="font-family: Symbol;">Å</span> (y * z) <span style="font-family: Symbol;">£</span> (x <span style="font-family: Symbol;">Å</span> y) * (x <span style="font-family: Symbol;">Å</span> z), we have
<br />x <span style="font-family: Symbol;">Å</span> (y * z) <span style="font-family: Symbol;">£</span> (x <span style="font-family: Symbol;">Å</span> y) * z.
<br />Hence the theorem.
<br />
<br /> <!--[if !supportLineBreakNewLine]--><b><nobr>
<br /><a name="6.3.3.4.2_Modular_Lattices">6.3.3.4.2 Modular Lattices</a>
<br /></nobr></b>
<br />In this section we shall define and discuss about the modular lattices.
<br /><i>A lattice L is said to be <b>modular
<br /></b>
<br />if for all x, y, z </i><i><span style="font-family: Symbol;">Î</span> L, x</i><i><span style="font-family: Symbol;">£</span> z </i><i><span style="font-family: Symbol;">Þ</span> x * (y </i><i><span style="font-family: Symbol;">Å</span> z) = (x * y) </i><i><span style="font-family: Symbol;">Å</span> z.
<br /></i>
<br /><b>Example 1</b>:
<br />
<br />(<i>P</i>(A), <span style="font-family: Symbol;">Ç</span> , <span style="font-family: Symbol;">È</span> ) is a modular lattice [ Proof left as an exercise].
<br />
<br /><b>
<br />Example 2</b>:
<br />
<br />The set of all normal subgroups of a group form a modular lattice. [Recall that a subgroup H of a group G is said to be normal if
<br />gHg<sup>-1</sup> = H, for all g <span style="font-family: Symbol;">Î</span> G]. It can be easily shown that M, the set of normal subgroups of a group G, with `set inclusion' relation is
<br />a poset.
<br />
<br />Now for any two normal subgroups H<sub>1</sub> and H<sub>2</sub> of G, we have
<br />H<sub>1</sub> * H<sub>2</sub> = inf (H<sub>1</sub>, H<sub>2</sub>) = H<sub>1</sub> <span style="font-family: Symbol;">Ç</span> H<sub>2</sub> and
<br />H<sub>1</sub> <span style="font-family: Symbol;">Å</span><sub> </sub>H<sub>2</sub> = sup (H<sub>1</sub>, H<sub>2</sub>) = H<sub>1</sub>H<sub>2</sub>.
<br />
<br />Since H<sub>1</sub> <span style="font-family: Symbol;">Ç</span> H<sub>2</sub> and H<sub>1</sub>H<sub>2</sub> are normal subgroups if both H<sub>1</sub> and H<sub>2</sub> are normal subgroups. Therefore, (M, <span style="font-family: Symbol;">Í</span> ) is a lattice ordered
<br />set. Hence (M, *, <span style="font-family: Symbol;">Å </span>) is an algebraic lattice. Let H<sub>1</sub>, H<sub>2</sub>, H<sub>3</sub>, <span style="font-family: Symbol;">Î</span> M such that H<sub>1</sub> <span style="font-family: Symbol;">Í</span> H<sub>3</sub>. Since in every lattice modular inequality
<br />holds good, we have H<sub>1</sub><!--[if gte vml 1]><v:shape id="_x0000_i1084" type="#_x0000_t75" alt="" style="'width:12.75pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image015.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image30.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image015.gif" shapes="_x0000_i1084" width="17" border="0" height="18" /><!--[endif]-->(H<sub>2</sub> * H<sub>3</sub>) <span style="font-family: Symbol;">Í</span> (H<sub>1</sub> <span style="font-family: Symbol;">Å</span> H<sub>2</sub>) * H<sub>3</sub> ………. (1)
<br />
<br />Now we shall prove that (H<sub>1</sub> <span style="font-family: Symbol;">Å</span> H<sub>2</sub>) * H<sub>3</sub> <span style="font-family: Symbol;">Í</span> H<sub>1</sub> <span style="font-family: Symbol;">Å</span> (H<sub>2</sub> * H<sub>3</sub>).
<br />Let a <span style="font-family: Symbol;">Î</span> (H<sub>1</sub> <span style="font-family: Symbol;">Å</span> H<sub>2</sub>) * H<sub>3
<br /></sub>i.e., a <span style="font-family: Symbol;">Î</span> (H<sub>1</sub> H<sub>2</sub>) <span style="font-family: Symbol;">Ç</span> H<sub>3
<br /><!--[if gte vml 1]><v:shape id="_x0000_i1085" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1085" width="20" border="0" height="16" /><!--[endif]--></sub>a <span style="font-family: Symbol;">Î</span> H<sub>1</sub> H<sub>2</sub> and a <span style="font-family: Symbol;">Î</span> H<sub>3
<br /><!--[if gte vml 1]><v:shape id="_x0000_i1086" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1086" width="20" border="0" height="16" /><!--[endif]--></sub>a = h<sub>1</sub>h<sub>2</sub> and a = h<sub>3</sub>, for some h<sub>1</sub> <span style="font-family: Symbol;">Î</span> H<sub>1</sub> ,h<sub>2</sub> <span style="font-family: Symbol;">Î</span> H<sub>2</sub> and h<sub>3</sub> <span style="font-family: Symbol;">Î</span> H<sub>3</sub>.
<br />Therefore, h<sub>1</sub>h<sub>2</sub> = h<sub>3
<br /></sub> <sub><!--[if gte vml 1]><v:shape id="_x0000_i1087" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1087" width="20" border="0" height="16" /><!--[endif]--></sub>h<sub>2</sub> = h<sub>1</sub><sup>-1</sup><sub> </sub>h<sub>3</sub> <span style="font-family: Symbol;">Î</span> H<sub>3</sub> [ h<sub>3</sub> <span style="font-family: Symbol;">Î</span> H<sub>3</sub> and h<sub>1</sub><span style="font-family: Symbol;">Î</span> H<sub>1</sub> <span style="font-family: Symbol;">Í</span> H<sub>3</sub> , therefore, h<sub>1</sub><sup>-1</sup> h<sub>3</sub><span style="font-family: Symbol;">Î</span> H<sub>3</sub>]
<br /> <sub><!--[if gte vml 1]><v:shape id="_x0000_i1088" type="#_x0000_t75" alt="" style="'width:15pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image21.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1088" width="20" border="0" height="16" /><!--[endif]--></sub>h<sub>2</sub><span style="font-family: Symbol;">Î</span> H<sub>2</sub> and h<sub>2</sub><span style="font-family: Symbol;">Î</span> H<sub>3</sub>
<br />Thus, h<sub>2</sub> <span style="font-family: Symbol;">Î</span> H<sub>2</sub> <span style="font-family: Symbol;">Ç</span> H<sub>3</sub> ……… (2)
<br />Since a = h<sub>1</sub>h<sub>2</sub>, we have a <span style="font-family: Symbol;">Î</span> H<sub>1</sub>(H<sub>2</sub> <span style="font-family: Symbol;">Ç</span> H<sub>3</sub>) (by (2))
<br />That is, a<span style="font-family: Symbol;">Î</span> H<sub>1</sub> <span style="font-family: Symbol;">Å</span><sub> </sub>(H<sub>2</sub> * H<sub>3</sub>)
<br />That is, (H<sub>1</sub> <span style="font-family: Symbol;">Å</span><sub> </sub>H<sub>2</sub>) * H<sub>3</sub> <span style="font-family: Symbol;">Í</span> H<sub>1</sub> <span style="font-family: Symbol;">Å</span><sub> </sub>(H<sub>2</sub> * H<sub>3</sub>) ………. (3)
<br />From (1) and (3) we have
<br />H<sub>1</sub> <span style="font-family: Symbol;">Å</span> (H<sub>2</sub> * H<sub>3</sub>) = (H<sub>1</sub> <span style="font-family: Symbol;">Å</span> H<sub>2</sub>) * H<sub>3</sub>.
<br />Hence (M, *,<span style="font-family: Symbol;">Å</span> ) is a modular lattice.
<br />
<br />
<br /><b>Theorem 3.4.2.1:
<br /></b>A lattice L is modular if and only if x, y, z <span style="font-family: Symbol;">Î</span> L , x <span style="font-family: Symbol;">Å</span> (y * (x <span style="font-family: Symbol;">Å</span> z)) = (x <span style="font-family: Symbol;">Å</span> y) * (x <span style="font-family: Symbol;">Å</span> z).
<br /><b>Proof :</b>
<br />Let (L, *, <span style="font-family: Symbol;">Å</span> ) be a modular lattice
<br />Then, if x <span style="font-family: Symbol;">£</span> z implies x <span style="font-family: Symbol;">Å</span> (y * z) = (x <span style="font-family: Symbol;">Å</span> y) * z ……… (1)
<br />But, for all x , z <span style="font-family: Symbol;">Î</span> L, x <span style="font-family: Symbol;">£</span> x <span style="font-family: Symbol;">Å</span> z,
<br />So, by (1) we have
<br />x <span style="font-family: Symbol;">Å</span> (y * (x <span style="font-family: Symbol;">Å</span> z)) = (x <span style="font-family: Symbol;">Å</span> y) * (x <span style="font-family: Symbol;">Å</span> z), for all x, y, z <span style="font-family: Symbol;">Î</span> L
<br />
<br />Conversely, suppose x <span style="font-family: Symbol;">Å</span> (y * (x <span style="font-family: Symbol;">Å</span> z)) = (x <span style="font-family: Symbol;">Å</span> y) * (x <span style="font-family: Symbol;">Å</span> z). ……… ( 2 )
<br />
<br />Then we shall prove that L is modular.
<br />If x <span style="font-family: Symbol;">£</span> z then x <span style="font-family: Symbol;">Å</span> z = z ………. (3)
<br />Substitute (3) in (2), we have if x <span style="font-family: Symbol;">£</span> z <span style="font-family: Symbol;">Þ</span> x <span style="font-family: Symbol;">Å</span> (y * z) = (x <span style="font-family: Symbol;">Å</span> y) * z
<br />Thus, (L, *, <span style="font-family: Symbol;">Å</span> ) is modular.
<br />
<br />
<br />If we have a characteristic result in terms of Hasse diagram for the modular lattice then it would effectively help in deciding a given
<br />lattice is modular or not. So, we shall prove the following lemma.
<br />
<br /><b>
<br />Lemma 3.4.2.1:
<br /></b>
<br />The "Pentagon Lattice" represented by the Hasse diagram given below is not modular.<b>
<br />
<br />Proof:
<br /></b>From the structure of the pentagon lattices we see that c <span style="font-family: Symbol;">£</span> a.
<br />Now c <span style="font-family: Symbol;">Å</span> (b * a) = c <span style="font-family: Symbol;">Å</span> 0 = c. On the other hand, (c <span style="font-family: Symbol;">Å</span> b) * a = 1 * a = a.
<br />Definitely c <span style="font-family: Symbol;">¹</span> a, thus, pentagon lattice is not a modular lattice.
<br />
<br />On the other hand, it can be proved that if any lattice whose substructure is isomorphic to a pentagon lattice cannot be a
<br />modular lattice. So, we have the following theorem.
<br /><b>
<br />Theorem 3.4.2.2:
<br /></b>A lattice L is modular if and only if none of its sublattice is isomorphic to the "pentagon lattice" [ for the detailed proof refer [2]]
<br /><b>
<br />Exercise 1:
<br /></b>Prove that the intervals [x, x <span style="font-family: Symbol;">Å</span><sub> </sub>y] and [ x * y, y] are isomorphic in a modular lattice. [ By interval [u ,v] we mean the set
<br />{ t <span style="font-family: Symbol;">Î </span>L / u<span style="font-family: Symbol;"> £ </span>t<span style="font-family: Symbol;"> £ </span>v }]
<br /><b>
<br />Exercise 2:
<br /></b>Prove that if a <span style="font-family: Symbol;">³</span> b and if there exists c <span style="font-family: Symbol;">Î</span> L with a <span style="font-family: Symbol;">Å </span>c = b <span style="font-family: Symbol;">Å </span>c and a * c = b * c then a = b.<o:p></o:p></p><div style="text-align: justify;"></div><p style="line-height: 150%; text-align: justify;"><b>
<br /><a name="6.3.4.3_Distributive_Lattices"><nobr>6.3.4.3 Distributive Lattices</nobr></a></b><nobr>In this section we will define and discuss distributive lattices.
<br /><i>
<br />A lattice (L, *, </i><i><span style="font-family: Symbol;">Å</span>) is called a <b>distributive lattice</b> if for any a, b, c </i><i><span style="font-family: Symbol;">Î</span> L,
<br /></i>a * (b <span style="font-family: Symbol;">Å</span> c) = (a * b) <span style="font-family: Symbol;">Å</span> (a * c)
<br />a <span style="font-family: Symbol;">Å</span> (b * c) = (a <span style="font-family: Symbol;">Å </span>b)*(a <span style="font-family: Symbol;">Å</span> c)
<br /><b>
<br />Example 1:
<br /></b>(<i>P</i>(A), <span style="font-family: Symbol;">Ç</span> , <span style="font-family: Symbol;">È</span> ) is a distributive lattice. [ For proof refer Section1.2]
<br /><b>
<br />Example 2:</b>
<br />Every totally ordered set is a distributive lattice.
<br /><b>Proof:</b>
<br />In a totally ordered set (T, <span style="font-family: Symbol;">£ </span>) for any two elements a, b in T, we have either a <span style="font-family: Symbol;">£ </span>b or b<span style="font-family: Symbol;">£ </span>a. Therefore, for any three elements a,
<br />b, c in T, the following are the possible situations in the structure of (T, <span style="font-family: Symbol;">£ </span>). <!--[if gte vml 1]><v:shape id="_x0000_i1091" type="#_x0000_t75" alt="" style="'width:249pt;height:192.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image017.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio18.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />All these possible choices can be covered in the following two cases.
<br />
<br /><b>Case 1: </b>a <span style="font-family: Symbol;">£</span><sub> </sub>b or a <span style="font-family: Symbol;">£</span><sub> </sub>c [ the first four choices ]
<br />
<br /><b>Case 2:</b> a <span style="font-family: Symbol;">³ </span>b and a <span style="font-family: Symbol;">³</span><sub> </sub>c [ the last two choices ]
<br />
<br />For the case 1, we have,
<br />a*(b <span style="font-family: Symbol;">Å</span> c) = a and (a * b) <span style="font-family: Symbol;">Å</span><sub> </sub>(a * c) = a <span style="font-family: Symbol;">Å</span><sub> </sub>a = a
<br />For the case 2, we have,
<br />a*(b <span style="font-family: Symbol;">Å</span><sub> </sub>c ) = b <span style="font-family: Symbol;">Å</span><sub> </sub>c and (a * b) <span style="font-family: Symbol;">Å</span><sub> </sub>(a * c) = b <span style="font-family: Symbol;">Å</span><sub> </sub>c
<br />Hence any totally ordered set (T, <span style="font-family: Symbol;">£</span><sub> </sub>) is a distributive lattice.
<br />
<br />
<br /><b>
<br />Example 3:
<br /></b>The set of all positive integers Z<sup>+</sup>, ordered by divisibility is a distributive lattice.
<br /><b>Proof:
<br /></b>We know that (Z<sup>+</sup>, <span style="font-family: Symbol;">½</span> ) is lattice, where x * y = GCD(x, y) and x <span style="font-family: Symbol;">Å</span><sub> </sub>y = LCM(x, y), for x, y <span style="font-family: Symbol;">Î</span> Z<sup>+</sup>.
<br />Since every positive integer can be expressed as product of powers of primes,
<br />
<br />Let x = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1092" type="#_x0000_t75" alt="" style="'width:37.5pt;height:34.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image018.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image32.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>, y = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1093" type="#_x0000_t75" alt="" style="'width:37.5pt;height:34.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image019.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image33.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>and z = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1094" type="#_x0000_t75" alt="" style="'width:36.75pt;height:34.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image020.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image34.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>.
<br />
<br />Note that x<sub>i </sub>, y<sub>i</sub> , z<sub>i </sub>may be zero also.
<br />Now , y * z =<sub> <!--[if gte vml 1]><v:shape id="_x0000_i1095" type="#_x0000_t75" alt="" style="'width:69.75pt;height:34.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image021.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image35.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>
<br />
<br />x <span style="font-family: Symbol;">Å</span> (y * z) = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1096" type="#_x0000_t75" alt="" style="'width:114pt;height:39.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image022.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image46.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>
<br />
<br /> = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1097" type="#_x0000_t75" alt="" style="'width:135.75pt;height:34.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image023.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image37.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>
<br />
<br /> = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1098" type="#_x0000_t75" alt="" style="'width:71.25pt;height:37.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image024.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image38.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>* <sub><!--[if gte vml 1]><v:shape id="_x0000_i1099" type="#_x0000_t75" alt="" style="'width:69.75pt;height:34.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image026.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image39.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></sub>
<br />
<br /> = (x <span style="font-family: Symbol;">Å</span><sub> </sub>y) * (x <span style="font-family: Symbol;">Å</span><sub> </sub>z)
<br /><b>
<br />Remark 1:</b>
<br />It is clear from the definition of distributive lattice that if a <span style="font-family: Symbol;">³</span> c then a * c = c and a <span style="font-family: Symbol;">Å</span><sub> </sub>c = a.
<br />
<br />Therefore, a * (b <span style="font-family: Symbol;">Å</span><sub> </sub>c) = (a * b) <span style="font-family: Symbol;">Å</span><sub> </sub>(a * c)
<br /> = (a * b) <span style="font-family: Symbol;">Å</span><sub> </sub>c.
<br />
<br />If a <span style="font-family: Symbol;">£ </span>c then a * c = a and a <span style="font-family: Symbol;">Å</span><sub> </sub>c = c.
<br />Therefore we have a <span style="font-family: Symbol;">Å</span><sub> </sub>(b * c) = (a <span style="font-family: Symbol;">Å</span><sub> </sub>b) * (a <span style="font-family: Symbol;">Å</span><sub> </sub>c)
<br /> = (a <span style="font-family: Symbol;">Å</span><sub> </sub>b) * c.
<br />
<br />Thus, every distributive lattice is modular.
<br />
<br />On the other hand every modular lattice need not be distributive.
<br />For example, the following modular lattice called <i>diamond lattice</i> is not distributive lattice.
<br /><b>
<br />Diamond Lattice:</b> <!--[if gte vml 1]><v:shape id="_x0000_i1100" type="#_x0000_t75" alt="" style="'width:168.75pt;height:165.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image027.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio19.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br />
<br />
<br />
<br /> For,<span style="font-family: Symbol;">Å</span><sub> </sub>(b * c) = a <span style="font-family: Symbol;">Å</span><sub> </sub>0 = a
<br />(a <span style="font-family: Symbol;">Å</span><sub> </sub>b) * (a <span style="font-family: Symbol;">Å</span><sub> </sub>c) = 1 * 1 = 1
<br />Therefore, a<span style="font-family: Symbol;">Å</span><sub> </sub>(b * c) <span style="font-family: Symbol;">¹</span> (a <span style="font-family: Symbol;">Å</span><sub> </sub>b) * (a <span style="font-family: Symbol;">Å</span><sub> </sub>c).
<br />Hence, a diamond lattice is not a distributive lattice.
<br />
<br />
<br />It is interesting to observe that if a lattice is not modular it cannot be a distributive lattice.
<br />It follows from the Theorem 3.4.2.2. and the above Remark1, we have the following theorem which will effectively decide the
<br />given lattice is distributive or not.
<br />
<br /><b>
<br />Theorem 3.4.3.1:</b>
<br />A lattice is distributive iff none of its sublattice is isomorphic to either the pentagon lattice or diamond lattice.
<br />[ For the proof refer [ 2 ] ]
<br /><b>
<br />Exercise:</b>
<br />Prove that the direct product of two distributive lattices is a distributive lattice.
<br /><b>
<br />Theorem 3.4.3.2:
<br /></b>Let (L, *, <span style="font-family: Symbol;">Å</span><sub> </sub>) be a distributive lattice. Then for any a, b, c <span style="font-family: Symbol;">Î</span> L,
<br />a * b = a * c and a <span style="font-family: Symbol;">Å</span><sub> </sub>b = a <span style="font-family: Symbol;">Å</span><sub> </sub>c <span style="font-family: Symbol;">Þ</span> b = c [cancellation law].
<br /><b>Proof:
<br /></b>(a * b) <span style="font-family: Symbol;">Å</span><sub> </sub>c = (a * c) <span style="font-family: Symbol;">Å</span><sub> </sub>c = c (since a * b = a * c and by absorption law, (a * c) <span style="font-family: Symbol;">Å</span><sub> </sub>c = c)
<br />
<br />Now, (a * b) <span style="font-family: Symbol;">Å</span><sub> </sub>c = (a <span style="font-family: Symbol;">Å</span><sub> </sub>c) * (b <span style="font-family: Symbol;">Å</span><sub> </sub>c) (by distributive law)
<br /> = (a <span style="font-family: Symbol;">Å</span><sub> </sub>b) * (b <span style="font-family: Symbol;">Å</span><sub> </sub>c) (since a <span style="font-family: Symbol;">Å</span><sub> </sub>c = a <span style="font-family: Symbol;">Å</span><sub> </sub>b)
<br /> = b <span style="font-family: Symbol;">Å</span><sub> </sub>(a * c) (by distributive law)
<br /> = b <span style="font-family: Symbol;">Å</span><sub> </sub>(a * b) (since a * c = a * b)
<br /> = b (by absorption law)
<br />Hence, b = c .
<br /> <!--[if !supportLineBreakNewLine]--></nobr><b><nobr><a name="6.3.4.4_Complemented_Lattices">3.4.4 Complemented Lattices</a></nobr></b><nobr>
<br />
<br />
<br />In this section we shall define complemented lattices and discuss briefly.
<br />
<br />
<br />In a lattice (L, *, <span style="font-family: Symbol;">Å </span>), the greatest element of the lattice is denoted by <b>1</b> and the least element is denoted by <b>0</b>.
<br />
<br />If a lattice (L, *,<span style="font-family: Symbol;">Å </span>) has <b>0</b> and <b>1</b>, then we have,
<br />x * <b>0</b> = <b>0</b>, x <span style="font-family: Symbol;">Å</span><sub> </sub><b>0</b> = x, x * <b>1</b> = x, x <span style="font-family: Symbol;">Å</span><sub> </sub><b>1</b> = <b>1</b>, for all x <span style="font-family: Symbol;">Î</span> L.
<br />
<br />
<br /><i>A lattice L with <b>0</b> and <b>1</b> is <b>complemented</b> if for each x in L there exists atleast one y </i><i><span style="font-family: Symbol;">Î</span> L such that x * y = <b>0</b> and
<br />x </i><span style="font-family: Symbol;">Å</span><i><sub> </sub>y = <b>1</b> and such element y is called complement of x.
<br /></i>
<br /><b>Note:
<br /></b>It is customary to denote complement of x by x'.
<br />
<br /><b>Remark 1:</b>
<br />It is clear that complement of <b>1</b> is <b>0</b> and vice versa.
<br /><b>
<br />Example 1:</b>
<br />Consider the lattice (<i>P</i>(A), <span style="font-family: Symbol;">Í</span> )
<br />Here <b>0</b> = <span style="font-family: Symbol;">f</span> and 1 = A.
<br />Then, for every S <span style="font-family: Symbol;">Î</span> <i>P</i>(A), that is, S <span style="font-family: Symbol;">Í</span> A, complement of S is A\ S, i.e. S'.
<br />
<br />
<br /><b>Example 2:</b>
<br />Consider the lattice L described in the Hasse diagram given below.
<br /> <!--[if gte vml 1]><v:shape id="_x0000_i1102" type="#_x0000_t75" alt="" style="'width:155.25pt;height:186.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image028.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Sectio20.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br /> Here, c does not have a complement.
<br />For, c * <b>0</b> = <b>0</b>, but, c <span style="font-family: Symbol;">Å</span> <b>0</b> = c.
<br />Therefore, <b>0</b> can not be a complement of c.
<br />
<br />Since a <span style="font-family: Symbol;">Å</span> c = c, therefore, a can not be a complement of c.
<br />
<br />Further, since c * b = c, b is not a complement of c.
<br />Also, c * <b>1</b> = c, <b>1</b> is not a complement of c.
<br />Hence, c does not have any complement in L.
<br />Therefore, L is not a complemented lattice.
<br />
<br /><b>Example 3:
<br /></b>In a totally ordered set, except <b>0</b> and <b>1</b>, all the other elements do not have complements.
<br />
<br /><b>Example 4:</b>
<br />Consider the lattice L described by the Hasse diagram given below. <!--[if gte vml 1]><v:shape id="_x0000_i1103" type="#_x0000_t75" alt="" style="'width:141.75pt;height:149.25pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image029.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image41.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]-->
<br /> Here, for c, we have, c * a = <b>0</b> and c <span style="font-family: Symbol;">Å</span> a =<b>1</b> and also, c * b = 0
<br />and c <span style="font-family: Symbol;">Å</span> b = <b>1</b>.
<br />Thus, c has a and b as its complements.
<br />From this example, it is clear that complement of an element in a complemented lattice need not be unique.
<br />
<br /><b>
<br />Theorem 3.4.5.1:</b>
<br />In a distributive lattice L with <b>0</b> and <b>1</b>, if a complement of an element exists then it is unique.
<br /><b>Proof:</b>
<br />Let L be a distributive lattice with <b>0</b> and <b>1</b>. Let a<span style="font-family: Symbol;">Î</span>L. Suppose a' and a" be two complement of a. Then by the definition of
<br />complement of an element, we have
<br />a * a' = <b>0 </b>and a <span style="font-family: Symbol;">Å</span> a' = <b>1</b>
<br />a <span style="font-family: Symbol;">Å</span> a" = <b>0</b> and a <span style="font-family: Symbol;">Å</span> a" = <b>1</b>
<br />
<br />Therefore, a * a' = a * a" and a <span style="font-family: Symbol;">Å</span> a' = a <span style="font-family: Symbol;">Å</span> a".
<br />Therefore by cancellation law in a distributive lattice, we have a' = a".
<br />Thus, complement of an element in a distributive lattice is unique.
<br />
<br /> <!--[if !supportLineBreakNewLine]--></nobr><span style="font-size: 13.5pt; line-height: 150%;"><nobr><a name="Exit_Quiz_Questions_:"><b>Exit Quiz Questions:</b></a>
<br /> <!--[if !supportLineBreakNewLine]-->
<br /> <!--[endif]--></nobr></span><o:p></o:p></p><div style="text-align: justify;"> <nobr> <ol start="1" type="1"><li class="MsoNormal" style="">Is the poset A = {2, 3, 6, 12, 24, 36, 72} under the relation of divisibility a lattice.<o:p></o:p></li><li class="MsoNormal" style="">If L<sub>1</sub> and L<sub>2</sub> are the lattices shown in the following figure, draw the Hasse diagram.of L<sub>1</sub><span style="font-family: Symbol;">´</span> L<sub>2</sub> with product partial order.<o:p></o:p></li></ol> <p class="MsoNormal"> <!--[if gte vml 1]><v:shape id="_x0000_i1105" type="#_x0000_t75" alt="" style="'width:90pt;height:96.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image030.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image42.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--><o:p></o:p></p> <ol start="3" type="1"><li class="MsoNormal" style="">Show that a subset of a totally ordered set is a sublattice.<o:p></o:p></li><li class="MsoNormal" style="">Find all the sublattices of D<sub>24</sub> that contains at least five elements.<o:p></o:p></li><li class="MsoNormal" style="">Shows that sublattice of a distributive lattice is distributive.<o:p></o:p></li><li class="MsoNormal" style="">Show that the lattice given below is not a distributive lattice but modular.<o:p></o:p></li></ol> <p class="MsoNormal"> <!--[if gte vml 1]><v:shape id="_x0000_i1106" type="#_x0000_t75" alt="" style="'width:104.25pt;height:99pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image031.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image43.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--><o:p></o:p></p> <ol start="7" type="1"><li class="MsoNormal" style="">Find the complement of each element of D<sub>42</sub>.<o:p></o:p></li><li class="MsoNormal" style="">Determine whether the lattice given below is distributive, complemented or both.<o:p></o:p></li></ol> <p class="MsoNormal"> <!--[if gte vml 1]><v:shape id="_x0000_i1107" type="#_x0000_t75" alt="" style="'width:36.75pt;height:123pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image032.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image44.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--><!--[if gte vml 1]><v:shape id="_x0000_i1108" type="#_x0000_t75" alt="" style="'width:2in;height:2in'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image033.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit6\Section6.3\Image\Image45.gif"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--><o:p></o:p></p> </nobr> </div><p style="text-align: justify;" class="MsoNormal"><o:p> </o:p></p> Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-25700066384937300762008-08-12T04:30:00.000-07:002008-12-23T01:55:02.554-08:00Partially Ordered Set and Hasse Diagram<b><nobr><a name="6.2.1 Relation and Poset">Relation and Poset</a>
<br /></nobr></b>
<br />In this section we define relation and special type of relation called reflexive, symmetric, anti-symmetric and transitive relation
<br />and we discuss examples on these relations. Further, we define partial ordering relation and posets and discuss some examples
<br /> of posets.
<br />
<br />Let A and B the non-empty sets. <i>A relation R from A to B is a subset of </i>A <span style="font-family:Symbol;">´</span> B. Relation from A to A is called relation on A.
<br /> If (a, b) <span style="font-family:Symbol;">Î</span> <i>R</i> then we write <i>aRb</i> and say that "a is in relation <i>R</i> to b". Also, if a is not in relation <i>R</i> to b, we write a<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Sectio1.gif" width="20" align="absmiddle" border="0" height="19" />b.
<br />
<br />
<br />
<br /><b>Example:</b>
<br />
<br />Let A = B = <b> Z<sup>+</sup></b>, the set of all positive integers. The relation <i>R</i> is defined on Z<sup>+</sup> in the following way <i>aRb</i> if and only if a divides b.
<br />So, 6 <i>R </i>18, but 3<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Sectio2.gif" width="20" align="absmiddle" border="0" height="19" />8
<br /><i> A relation R on a set A is <b>reflexive</b> if (a, a) for every a<span style="font-family:Symbol;">Î</span> A, that is, if aRa for all a<span style="font-family:Symbol;">Î</span> A.</i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image14.gif" width="12" height="22" />
<br /><b>
<br />Example:</b>
<br />
<br />Let A= {1,2,3} and let <i>R</i> = {(1,1), (2,2), (3,3), (1,2) (1,3)}
<br />Then <i>R</i> is reflexive, while <i>R'</i> = {(1,1), (2,2), (2,1), (1,2)} is not a reflexive, for (3,3) <span style="font-family:Symbol;">Ï</span> <i>R'</i>, i.e., 3<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image15.gif" width="20" height="24" /></sub>3.
<br />
<br /><i>A relation R, on a set A, is <b>anti-symmetric</b> if whenever aRb and bRa, then a = b.</i> <i>That is, R is anti-symmetric whenever
<br />a <span style="font-family:Symbol;">¹</span> b we must have either a<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Sectio3.gif" width="20" align="absmiddle" border="0" height="19" />b or b<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Sectio4.gif" width="20" align="absmiddle" border="0" height="19" />a.
<br /></i><b>
<br />
<br />
<br />Example 1:
<br />
<br /></b>Let X be a set and let A <span style="font-family:Symbol;">Í</span> X and B <span style="font-family:Symbol;">Í</span> X. Then from the Section 1.1, it is clear that A = B. Therefore, "<span style="font-family:Symbol;">Í</span> " is anti-symmetric
<br /> on the set of subsets of X.
<br /><b>
<br />Example 2:
<br />
<br /></b>Let A=Z<sup>+</sup>, the set of all positive integer. Define <i>R </i>on A by <i>aRb</i> if and only if<i> a<span style="font-family:Symbol;"> </span></i><span style="font-family:Symbol;">½</span><i> b</i> that is <i>"a divides b".</i>
<br />Then, if <i>a R b</i> and <i>b R a,</i> i.e., <i>a</i><span style="font-family:Symbol;">½</span><i>b</i> and <i>b</i><span style="font-family:Symbol;">½</span><i>a</i>, then there exist integers c, d <span style="font-family:Symbol;">Î</span> Z<sup>+</sup> such that <i>b = ca</i> and <i>a = db.</i>
<br /> Therefore, <i>b = cbd</i>.
<br />So,<i> 1 = cd. </i> Therefore <i>c = d = 1.</i>
<br />Hence<i> a = b.
<br /></i>Therefore, <i>R</i> is anti-symmetric on Z<sup>+
<br /></sup>
<br /><b>Example 3:</b>
<br />
<br />Let A = Z, the set of integers and let <i>R = {(a, b) <span style="font-family:Symbol;">Î</span> A <span style="font-family:Symbol;">´</span> A / a <> i.e., <i>R</i> is usual <i>"less than".
<br /></i> If a <span style="font-family:Symbol;">¹</span> b, then either a < face="Symbol">¹</span> b then either b<span style="font-size:85%;"><sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image16.gif" width="17" align="absmiddle" height="23" /></sub></span> a or a<span style="font-size:85%;"><sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image16.gif" width="17" align="absmiddle" height="23" /></sub></span>b.
<br />Hence,<b>if a <span style="font-family:Symbol;">¹</span> b then either a<span style="font-size:85%;"> <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image16.gif" width="15" align="absmiddle" height="22" /></sub></span>b or b<span style="font-size:85%;"><sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image16.gif" width="15" align="absmiddle" height="21" /></sub></span>a is true</b>.
<br />Therefore, "<", usual "less than", is anti-symmetric.
<br />
<br />[Note that in the anti-symmetric relation symmetryness will happen only with equal elements and symmetryness will never
<br /> happen between the unequal elements].
<br /><i>
<br />We say that a relation R on a set A is <b>transitive</b> if aRb and bRc then aRc.
<br /></i>
<br /><b>Example:</b>
<br />
<br />Let A={1,2,3,4} and let R={(1,2), (2,1), (1,1), (2,2), (2,3), (2,3)}. Then R is transitive.
<br />Let R={(1,2) (1,3), (4,2)}. Then R is also transitive, because we cannot find elements a, b, and c in A such that aRb and bRc,
<br /> but a <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Sectio5.gif" width="20" align="absmiddle" border="0" height="19" />c. So, we conclude that R is transitive.
<br />
<br /><i>A relation R on a set A is called a <b>partial order relation</b> or <b>partial ordering</b> if R is reflexive, anti-symmetric and
<br /> transitive.
<br />
<br />A non-empty set A together with a partial order relation R, (A, R), is called <b>partially ordered set</b> or <b>poset</b>.
<br /></i>
<br />Partial order relations are "hierarchical relations", usually we write <span style="font-family:Symbol;">£</span> instead of R for partial ordering.
<br />Thus, (A, <span style="font-family:Symbol;">£</span> ) denote a partially ordered set or poset.
<br />
<br /><b>Example 1:</b>
<br />
<br />Let A be a non empty set. Consider the power set <i>P</i>(A), the set of subsets of A, together with the relation set inclusion, "<span style="font-family:Symbol;">Í</span> ".
<br /> Then (<i>P</i>(A), <span style="font-family:Symbol;">Í</span> ) is a poset.
<br />
<br />For, S <span style="font-family:Symbol;">Í</span> S, for all S <span style="font-family:Symbol;">Í</span> <i>P</i>(A).
<br />Therefore, <span style="font-family:Symbol;">Í</span> is reflexive.
<br />If S <span style="font-family:Symbol;">Í</span> T and T <span style="font-family:Symbol;">Í</span> S, for S, T <span style="font-family:Symbol;">Í</span> <i>P</i>(A),
<br />then S = T (since S = T if and only if S <span style="font-family:Symbol;">Í</span> T and T <span style="font-family:Symbol;">Í</span> S).
<br />Therefore, <span style="font-family:Symbol;">Í</span> is anti-symmetric.
<br />Also, if S <span style="font-family:Symbol;">Í</span> T and T <span style="font-family:Symbol;">Í</span> U, then by definition of <span style="font-family:Symbol;">Í</span> , it is clear that S <span style="font-family:Symbol;">Í</span> U.
<br />Therefore, <span style="font-family:Symbol;">Í</span> is transitive.
<br />Hence, (<i>P</i>(A), <span style="font-family:Symbol;">Í</span> ) is a partially ordered set.
<br /><b>
<br />Example 2:</b>
<br />
<br />Let A = Z<sup>+</sup> and let a <span style="font-family:Symbol;">£</span> b if and only if a<span style="font-family:Symbol;">½</span>b then (A, <span style="font-family:Symbol;">£</span> ) is a poset.
<br />For, since a = 1.a ,<span style="font-family:Symbol;">"</span> a <span style="font-family:Symbol;">Î</span> Z<sup>+</sup>, i.e., a<span style="font-family:Symbol;">½</span>a, <span style="font-family:Symbol;">"</span> a <span style="font-family:Symbol;">Î</span> Z<sup>+.</sup>
<br />Therefore "<span style="font-family:Symbol;">£</span> " is reflexive.
<br />If a<span style="font-family:Symbol;">½</span>b and b<span style="font-family:Symbol;">½</span>c, then there exists d<sub>1</sub> and d<sub>2</sub> in Z<sup>+</sup>, such that b = d<sub>1</sub>a and c = d<sub>2</sub>b.
<br />So, we have c = d<sub>2</sub>d<sub>1</sub>a.
<br />As d<sub>1</sub>, d<sub>2</sub> <span style="font-family:Symbol;">Î</span> Z<sup>+</sup>, d<sub>2</sub>d<sub>1</sub> <span style="font-family:Symbol;">Î</span> Z<sup>+
<br /></sup>Then a<span style="font-family:Symbol;">½</span>c.
<br />Hence, " <span style="font-family:Symbol;">£</span> " is transitive.
<br />Already we have seen that " <span style="font-family:Symbol;">£</span> " is anti-symmetric.
<br />Thus, " <span style="font-family:Symbol;">£</span> " is partial ordering.
<br />Hence, (A, <span style="font-family:Symbol;">£</span> ) is a poset.
<br /><b>
<br />Example 3:
<br /></b>(R, <span style="font-family:Symbol;">£</span> ) is a poset, where R is the set of all real number and " <span style="font-family:Symbol;">£</span> " is "usual less than or equal to " (Prove!).
<br /><b>
<br />Example 4:
<br /></b>(<i>P</i>(S), <span style="font-family:Symbol;">Ì</span> ) is not a poset, for, the relation <span style="font-family:Symbol;">Ì</span> is anti-symmetric and transitive but not reflexive so it is not a poset.
<br /><b>
<br />Example 5:
<br /></b>Let S be the set of all subgroups of a group G. Then (S, <span style="font-family:Symbol;">Í</span> ) is a poset (Prove!)
<br /><b>
<br />Example 6:
<br /></b>Let S be the set of all normal subgroups of G. then (S, <span style="font-family:Symbol;">Í</span> ) is a poset (Prove!).
<br /><b>
<br />Example 7:
<br /></b>Let S = {( i<span style="font-size:100%;"><sub>1</sub></span>, i<span style="font-size:100%;"><sub>2</sub></span>,…,i<span style="font-size:100%;"><sub>n</sub></span>) / i<span style="font-size:100%;"><sub>r</sub></span><span style="font-size:180%;"><sub> </sub></span>= 0 or 1, 1 <span style="font-family:Symbol;">£</span> r <span style="font-family:Symbol;">£</span> n}
<br />Define (i<span style="font-size:100%;"><sub>1</sub></span>,i<span style="font-size:100%;"><sub>2</sub></span>,…,i<span style="font-size:100%;"><sub>n</sub></span>) <i>R</i> ( j<span style="font-size:100%;"><sub>1</sub></span>,j<span style="font-size:100%;"><sub>2</sub></span><span style="font-size:130%;">,…</span>,j<span style="font-size:100%;"><sub>n</sub></span><sub> </sub>) if and only if i<sub>1 </sub> <span style="font-family:Symbol;">£</span> <sub> </sub>j<sub>1</sub>, i<sub>2 </sub> <span style="font-family:Symbol;">£</span> <sub> </sub>j<sub>2</sub>,…,i<span style="font-size:100%;"><sub>n</sub></span><sub> </sub><span style="font-family:Symbol;">£</span><sub> </sub>j<span style="font-size:100%;"><sub>n</sub></span>.
<br />Then (S,<i>R</i>) is a poset (Prove!).
<br /><span style="font-size:85%;"> </span><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Section6.2.htm#6.2%20Partially%20Ordered%20Set%20and%20Hasse%20Diagram">Back to top</a> </span></p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.1/Image/Sectio14.gif" width="778" border="0" height="10" /> <nobr>
<br /><a name="6.2.2 Hasse Diagram">
<br />6.2.2 Hasse Diagram</a>
<br /></nobr></b>
<br />In this section we discuss the diagrammatic representation of a poset.
<br />
<br />In a poset (A, <span style="font-family:Symbol;">£</span> ), if a <span style="font-family:Symbol;">£</span> b and a <span style="font-family:Symbol;">¹ </span>b then we write a <>in a poset (A, <span style="font-family:Symbol;">£</span> ), we say that a is a <b>cover</b> of b if a < b
<br /> and there exists no u such that a <>
<br /><i>
<br />A finite poset can be represented graphically, such a diagram of a poset is called <b>Hasse diagram</b></i>. We represent the
<br /> elements of A by points in the plane. If <i>b</i> is a cover of <i>a</i> then we place <i>b</i> above <i>a</i> and connect the two points <i>a</i> and <i>b</i> by a
<br /> straight line (actually directing from <i>a</i> to <i>b</i>). Thus, if a < b if and only if there is a ascending broken line (a path) connecting a to b.
<br /> If no line connecting a and b and a <span style="font-family:Symbol;">¹</span> b, then a and b are not related or not comparable, that is, we have neither a <span style="font-family:Symbol;">£</span> b nor b <span style="font-family:Symbol;">£</span> a.
<br /><b>
<br />
<br />
<br />Example</b>:
<br />
<br />1. Hasse diagram of the poset ({1,2,3,4,5}, <span style="font-family:Symbol;">£</span> ), where <span style="font-family:Symbol;">£</span> is "usual less than or equal to" is given below.
<br />
<br /> 2. Consider the poset (<i>P</i>(A<span style="font-size:100%;"><sub>3</sub></span>), <span style="font-family:Symbol;">Í</span> ), where A<span style="font-size:100%;"><sub>3</sub></span> = {a, b, c}. The Hasse diagram of (<i>P</i>(A<span style="font-size:100%;"><sub>3</sub></span>), <span style="font-family:Symbol;">Í</span> ) is given below :
<br /> te that actually all the lines should have upward directions. Since all the lines are having only one direction, it is a convention
<br /> to draw without direction in the lines. Thus, in the Hasse diagram every path has only upward direction, i.e., there cannot be any
<br /> path connecting top to bottom in the downward direction.
<br /> In the above example we see that {a} and {a, b, c} are related, since there is an ascending path from {a} to {a, b, c}, while {a}
<br /> and {b} are not related or not comparable, since there is no ascending path connecting {a} and {b}.
<br />
<br />3. Let D<span style="font-size:100%;"><sub>n</sub></span> denote the set of all positive divisiors of a positive integer n.
<br /> Hasse diagram of the poset (D<span style="font-size:100%;"><sub>12</sub></span>, <span style="font-family:Symbol;">½</span>) is given below.
<br />
<br />
<br />
<br />Conversely, if we have a Hasse diagram then we can describe the poset, that is, we can describe the partial order relation.
<br /> For example, consider the Hasse diagram given below :
<br />
<br />
<br />
<br />Then the set is {a, b, c, d, e, f, g} and the relation <span style="font-family:Symbol;">£</span> is {(a, a) (a, c) (a, d), (a,e), (a, f), (a, g), (b, b), (b, c), (b, d), (g, e), (g, f),
<br /> (b, g), (c, c), (c, d), (c, e), (c, f), (c, g), (d, d), (d, e), (d, f), (d, g), (e, e), (e, g), (f, f), (f, g), (g, g)}.
<br />
<br />One can check that <span style="font-family:Symbol;">£</span> is indeed partial ordering.
<br />
<br /> So,<b><i> we conclude that for the finite poset it is enough to just give the Hasse diagram to describe the poset.</i></b>
<br /> <b>
<br />Remark:
<br /></b>
<br />From the above examples of the posets, we observe that in a poset it is not necessary that all the pair of elements are related or
<br />comparable, that is, in a poset not necessarily all the elements are ordered. That is the reason we call such sets (posets) as
<br /> partially ordered set.
<br /><span style="font-size:85%;"> </span></p><p style="line-height: 150%;" align="left"><b>
<br /><nobr>
<br /><a name="6.2.3 Totally Ordered Set and Dual Poset">6.2.3 Totally Ordered Set and Dual Poset</a>
<br /></nobr> </b> <nobr>
<br />
<br />In this section we define totally ordered set and dual poset and discuss about them.
<br /><i>
<br />A partial order relation <span style="font-family:Symbol;">£</span> on a set A is called a <b>total order</b> (or linear order) if, for each a, b <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image18.gif" width="13" height="13" /></sub> A either a <span style="font-family:Symbol;">£</span> b or b <span style="font-family:Symbol;">£</span> a.
<br /> If <span style="font-family:Symbol;">£</span> is a totally order on a set A, then (A, <span style="font-family:Symbol;">£</span> ) is called <b>totally ordered set</b> or <b>linearly ordered set</b> or <b>chain</b>.
<br /></i><b>
<br />Example 1:</b>
<br />
<br />The poset (N, <span style="font-family:Symbol;">£</span> ), where N is the set of all natural numbers and <span style="font-family:Symbol;">£</span> is the usual "less than or equal to", is a totally ordered set, since
<br /> for any two a, b <span style="font-family:Symbol;">Î</span> N, we have either a <span style="font-family:Symbol;">£</span> b or b <span style="font-family:Symbol;">£</span> a.
<br />
<br />In fact, ( Z, <span style="font-family:Symbol;">£</span> ), ( Q, <span style="font-family:Symbol;">£</span> ) ( R, <span style="font-family:Symbol;">£</span> ), are all totally ordered set, where Z, the set of all integers, Q, the set of all rational numbers and
<br /> R, the set of all real number and <span style="font-family:Symbol;">£</span> is usual less than or equal to. Also, any finite subset of either Z or Q or R with usual "less than
<br /> or equal to" are also totally ordered sets.
<br /><b>
<br />Example 2:
<br /></b>( D<sub>30</sub>, <span style="font-family:Symbol;">½</span>) is not a totally ordered set (Prove !)
<br />
<br /><b>
<br />Exercise:
<br />
<br /></b>Let L be the set of complex numbers z = x + iy, where x and y are rationals.
<br />Define a partial order " <span style="font-family:Symbol;">£</span> " on L by: x<span style="font-size:100%;"><sub>1</sub></span> + iy<span style="font-size:100%;"><sub>1 </sub></span><span style="font-family:Symbol;">£</span><span style="font-size:130%;"><sub> </sub></span>x<span style="font-size:100%;"><sub>2</sub></span> + iy<span style="font-size:100%;"><sub>2</sub></span> if and only if y<span style="font-size:100%;"><sub>1</sub></span> <span style="font-family:Symbol;">£</span> y<span style="font-size:100%;"><sub>2</sub></span>.
<br />
<br />Is ( L, <span style="font-family:Symbol;">£</span> ) a totally ordered set. If not, what is the additional condition needed in order to make ( L, <span style="font-family:Symbol;">£</span> ) into a chain?
<br /><b>
<br />Remark 1:
<br /></b>From the definition of totally ordered set, it is clear that any two elements in a totally ordered set are related or comparable. That
<br /> is, in a totally ordered set all the elements are ordered or related. That is the reason we call such sets as totally ordered set.
<br />
<br /><i>
<br />If R is a relation from A to B then R<sup>-1</sup> defined by (a, b)<span style="font-family:Symbol;">Î</span> R<sup><span style="font-family:p,Times New Roman;">-1</span></sup> if and only if (b, a)<span style="font-family:Symbol;">Î</span> R, is a relation from B to A, called the <b>
<br />converse relation</b> of R.
<br /></i><b>
<br />Remark 2:
<br /></b>If (A, <i>R</i>) is a partially ordered set then (A, <i>R<sup>-1</sup></i>) is a partial ordered set.
<br /><b>
<br />Proof:
<br /></b>Let (A, <i>R</i>) be a poset. Then <i>R</i> is a partial ordering on A. Therefore, <i>R</i> is reflexive, anti-symmetric and transitive.
<br />Now we shall prove that <i>R<sup><span style="font-family:p,Times New Roman;">-1</span></sup></i> is reflexive, anti-symmetric and transitive.
<br />Since <i>R</i> is reflexive, we have, (a, a) <span style="font-family:Symbol;">Î</span> <i>R</i> ,for all a <span style="font-family:Symbol;">Î</span> A.
<br />Therefore, (a, a) <span style="font-family:Symbol;">Î</span> <i>R<sup>-1</sup>,<sup> </sup></i>for all a <span style="font-family:Symbol;">Î</span> A also.
<br />Thus, <i>R<sup>-1</sup></i><sup> </sup>is reflexive.
<br />If a<i>R<sup>-1</sup> </i>b and b<i>R<sup>-1</sup></i>a, then by definition of <i>R<sup>-1</sup></i> we have b<i>R</i>a and a<i>R</i>b.
<br />Since <i>R</i> is anti-symmetric, we have a = b. Therefore, <i>R<sup>-1</sup></i> is anti-symmetric.
<br />Let a<i>R<sup>-1</sup> </i>b and b<i>R<sup>-1</sup></i>c. Then by definition of <i>R<sup>-1</sup></i>, we have b<i>R</i>a and c<i>R</i>b.
<br />Since R is transitive, we have c<i>R</i>a. Therefore, a<i>R<sup>-1</sup></i>c. Thus,<i> R<sup>-1</sup></i> is transitive.
<br />Hence (A, <i>R<sup>-1</sup></i>) is a partially ordered set.
<br /><b>
<br />Note:
<br /></b>
<br />Since we use <span style="font-family:Symbol;">£</span> for denoting partial ordering we denote <span style="font-family:Symbol;">³</span> for its converse. Thus, if (A, <span style="font-family:Symbol;">£</span> ) is a partial ordered set then (A, <span style="font-family:Symbol;">³</span> ) is
<br /> also partial ordered set. The poset (A, <span style="font-family:Symbol;">³</span> ) is called dual poset to the poset (A, <span style="font-family:Symbol;">£</span> ).
<br /><b>
<br />Remark 3: (Duality Principle)</b>
<br />
<br />Every statement or formula or expression on an ordered set (A, <span style="font-family:Symbol;">£</span> ) remains correct if everywhere in the statement the relation <span style="font-family:Symbol;">£</span>
<br /> is replaced by its converse relation <span style="font-family:Symbol;">³</span> .
<br /></nobr> <span style="font-size:85%;"> </span></p><p style="line-height: 150%;" align="right"><span style="font-size:85%;">
<br /></span></p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Section6.2.htm#6.2%20Partially%20Ordered%20Set%20and%20Hasse%20Diagram">Back to top</a> </span></p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.1/Image/Sectio14.gif" width="778" border="0" height="10" />
<br /><nobr>
<br /><a name="6.2.4 Extremal Elements of Partially Ordered Sets">6.2.4 Extremal Elements of Partially Ordered Sets</a>
<br /></nobr></b><nobr>
<br />In this section we discuss about the extremal elements of a poset. Let (A, <span style="font-family:Symbol;">£</span> ) be a poset. <i>An element a<span style="font-family:Symbol;">Î</span> A is called a <b>maximal
<br /> element</b> of A if there is no element c in A such that </i> <i>a < face="Symbol">Î</span> A is called a <b>minimal element</b> of A if there is no
<br /> element c in A such that c < face="Symbol">Î</span> A is called a <b>greatest element</b> of A if x <span style="font-family:Symbol;">£</span> a, for all x<span style="font-family:Symbol;">Î</span> A. Similarly, an
<br /> element a<span style="font-family:Symbol;">Î</span> A is called a <b>least element</b> of A if a <span style="font-family:Symbol;">£</span> x, for all x<span style="font-family:Symbol;">Î</span> A.
<br /></i><b>
<br />Example:
<br /></b>
<br />Consider the following poset represented by the following Hasse diagram
<br />
<br />In this poset, a, f and i are minimal elements c, h, k are maximal elements, there is a no greatest element and there is no least
<br />element.
<br />
<br />Consider the following posets represented by Hasse diagrams.
<br />
<br />
<br />In the poset (i), a is the least and minimal element and d is the greatest and maximal element.
<br />In the poset (ii), a is the least and minimal element and d and e are maximal elements but there is no greatest element.
<br />
<br /><b>Remark:
<br /></b>
<br />In a poset, least element or greatest element need not always exist. It is clear that least element is a minimal element and greatest
<br /> element is a maximal element but not conversely.
<br />
<br /><b>
<br />Exercise:</b>
<br /><ol><li> Let (A, <span style="font-family:Symbol;">£</span> <sub> </sub>) be a finite non empty poset. Then prove that A has at least one maximal element and at least one minimal
<br /> element.
<br /> </li><li> Let (A, <span style="font-family:Symbol;">£ </span>) be a poset. Then prove that if least element or greatest element exist then they are unique. </li></ol>
<br />
<br />Let (A, <span style="font-family:Symbol;">£</span> ) be a poset and B <span style="font-family:Symbol;">Í </span>A.
<br /><ol type="i"><li> <i>a <span style="font-family:Symbol;">Î</span> A is called an <b>upper bound </b>of B if and only if b </i><span style="font-family:Symbol;">£</span> <i> a, for all b <span style="font-family:Symbol;">Î</span> B.
<br /></i></li><li> <i> a <span style="font-family:Symbol;">Î</span> A is called a <b>lower bound</b> of B if and only if a </i><span style="font-family:Symbol;">£</span> <i> b, for all b <span style="font-family:Symbol;">Î</span> B.
<br /> </i> </li><li> <i> An lower bound g of B is called a <b>greatest lower bound or infimum </b>if and only if h <span style="font-family:Symbol;">£</span> g for every lower bound h of
<br /> B in A and it is denoted by <b>inf B</b> or <b>GLB of B.
<br /> </b></i></li><li> <i> An upper bound v of B is called least upper bound or <b>superimum </b>if and only if v <span style="font-family:Symbol;">£</span> <sub> </sub>u for all upper bound u of B in A
<br /> and it is denoted by <b>sup B</b> or <b>LUB of B.</b></i></li></ol>
<br /><b>Example:</b>
<br />
<br />Consider the poset (D<sub>30</sub> ,<span style="font-family:Symbol;">½</span>), i..e ({30, 15, 10, 6, 5, 3, 2, 1}, <span style="font-family:Symbol;">½</span> ).
<br />
<br />Let B = {2, 3, 6}. Then inf B = 1, upper bounds of B are 6 and 30 but supB = 6.
<br />
<br /><b>Exercise:</b>
<br />
<br />Let (A, <span style="font-family:Symbol;">£</span> ) be a poset and let B <span style="font-family:Symbol;">Í</span> A. Prove that if <b>inf B</b> or <b>sup B</b> exist then they are unique.
<br /></nobr> <span style="font-size:85%;"> </span></p><p style="line-height: 150%;" align="left"><b><span style="font-size:130%;">
<br /><nobr>
<br /><a name="Exist Quiz Questions:">Exit Quiz Questions:</a>
<br /></nobr></span></b> </p><ol><li> On the set A = {a, b, c}, find all partial orders <span style="font-family:Symbol;">£</span> in which a <span style="font-family:Symbol;">£</span> b?
<br /> </li><li> If (A, <span style="font-family:Symbol;">£</span> ) is a poset and A' is a subset of A. check whether (A', <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image20.gif" width="12" height="16" /></sub>' ) is also a poset, where <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.2/Image/Image20.gif" width="12" height="16" /></sub>' is the restriction of <span style="font-family:Symbol;">£</span> to A.
<br /> </li><li> Construct the Hasse diagram of (D<sub>30</sub> ,<span style="font-family:Symbol;">½</span>) and (P(A<sub>3</sub>), <span style="font-family:Symbol;">£ </span>), where A<sub>3</sub> = {1,2,3} Do they have structural similarity?
<br /> </li><li> Find ( i ) all the lower bound of B.
<br /> ( ii ) all the upper bound of B.
<br /> ( iii ) the least upper bound of B.
<br /> ( iv ) the greatest lower bound of B, where B = {c, d, e} in the poset represented by the following poset.
<br />
<br /></li><li> Whether every finite poset has maximal element? If yes, justify your answer</li></ol>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1tag:blogger.com,1999:blog-7399621945608861143.post-27765295665226810112008-08-12T04:29:00.002-07:002008-12-23T01:55:02.554-08:00Sets and Motivation for Boolean Algebra<b><a name="6.1.1 Sets"><span style="font-size:100%;">Sets</span></a><br /></b><br /><span style="font-size:100%;">We devote this section to brief some set theoretic notions, which will play an essential role in the lattice theory.</span><br /><b><i><br /><span style="font-size:100%;">Set</span></i></b><span style="font-size:100%;"><i> is a collection of well-defined objects</i>. <i>An object belonging to a set is called <b>member</b> or <b>element</b> of the <b>set</b>. </i></span><br /><br /><span style="font-size:100%;">In most cases, set will be defined by means of a characteristic property of the objects belonging to the set. That is, for a given<br />property P(x), let {x : P(x)} denote the set of all objects x such that P(x) is true. <i>A set with no member is said to be an <b>empty<br /> set</b>.</i> We use upper case letters such as A, B, C, etc., to denote sets and lower case letters such as a, b, c, x, y, z, etc, to denote<br /> members of the sets. We denote the fact that x is an element of the set S by x <span style="font-family:Symbol;">Î</span> S, while x is not an element of S by x <span style="font-family:Symbol;">Ï</span> S.</span><br /><i><br /><span style="font-size:100%;">Two sets <b> A and B are equal</b> if and only if A and B have the same members.</span></i> <span style="font-size:100%;"> Equality of A and B is denoted by A=B. </span><i><span style="font-size:100%;">We<br /> say that <b>A is a subset of B</b> if and only if every member of A is also a member of B</span></i><span style="font-size:100%;">. We write A <span style="font-family:Symbol;">Í</span> B as an abbreviation for <b><br />A is a subset of B</b>.<br /><br />It is interesting to observe that, for all sets A, B and C, we have<br />(i) A <span style="font-family:Symbol;">Í</span> A [Reflexive]<br />(ii) A <span style="font-family:Symbol;">Í</span> B and B<span style="font-family:Symbol;">Í</span> A if and only if A=B<br />(iii) If A <span style="font-family:Symbol;">Í</span> B, B <span style="font-family:Symbol;">Í</span> C then A<span style="font-family:Symbol;">Í</span> C [transitive]<br /></span><br /><i><br /><span style="font-size:100%;">P(A) denote the set of all subset of A, we shall call P(A), <b>the power set of A.<br /></b></span></i><span style="font-size:100%;">That is, <i>P</i>(A) = {B : B <span style="font-family:Symbol;">Í</span> A}</span><br /><b><br /><span style="font-size:100%;">Example 1:</span></b><span style="font-size:100%;"><br />Let A = {1,2,3}<br />Then <i>P</i>(A) = {<span style="font-family:Symbol;">f</span> , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}.</span><br /><br /><span style="font-size:100%;">Let A be a finite set of n elements, say A= {a<sub>1</sub>, a<sub>2</sub>, …, a<sub>n</sub>}, then <i>P</i>(A) consists of <span style="font-family:Symbol;">f</span> , the n sets {a<sub>i</sub>} containing single element,<br /> nC<sub>2</sub> sets {a<sub>i</sub>, a<sub>j</sub> / i</span><span style="font-size:100%;"> </span><span style="font-family:Symbol;">¹ </span><span style="font-size:100%;">j}containing two elements, etc. In general P(A) contains nC<sub>i</sub> subsets containing i distinct elements of A, for<br /> 1<span style="font-family:Symbol;">£</span> i <span style="font-family:Symbol;">£</span> n. Therefore the number of elements in <i>P</i>(A) is 1+ nC<sub>1</sub> + nC<sub>2</sub> + ……….+ nC<sub>n</sub> = (1+1)<sup>n</sup> = 2<sup>n</sup>.</span><br /><br /> <p style="line-height: 150%;" align="right"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.1/Section6.1.htm#6.1%20Sets%20and%20Motivation%20for%20Boolean%20Algebra"><span style="font-size:85%;">Back to top</span></a> </p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.1/Image/Sectio12.gif" width="767" border="0" height="10" /> <nobr><br /><a name="6.1.2 Algebra of Sets"><span style="font-size:100%;"><br />6.1.2 Algebra of Sets</span></a><br /></nobr></b><br /><span style="font-size:100%;">In this section we define, union of two sets, intersection of two sets, complement of a set and we brief some of their basic<br />properties and their operations.<br /><i>Given sets A and B, their <b>union A </b></i><b> <span style="font-family:Symbol;">È</span> <i> B</i></b> <i> consists of all elements in A or B or both</i>.</span><br /><br /><span style="font-size:100%;">That is, A<span style="font-family:Symbol;">È</span> B = {x / x <span style="font-family:Symbol;">Î</span> A or x <span style="font-family:Symbol;">Î</span> B}.</span><br /><br /><span style="font-size:100%;">It is interesting to observe that the union operation on sets has properties:</span><br /></p><ol type="i"><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">È</span>A = A (Idempotent)<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">È </span>B = B <span style="font-family:Symbol;">È</span>A (Commutative)<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">È </span><span style="font-family:Symbol;font-size:100%;">f</span> = A<br /> </span></li><li> <span style="font-size:100%;">(A <span style="font-family:Symbol;">È </span>B) <span style="font-family:Symbol;">È </span>C = A <span style="font-family:Symbol;">È </span>(B <span style="font-family:Symbol;">È</span> C) (Associative)<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">È </span>B = B if and only if A<span style="font-family:Symbol;">Í</span> B<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">Í</span> (A <span style="font-family:Symbol;">È </span>B) and B <span style="font-family:Symbol;">Í</span> (A<sub> </sub><span style="font-family:Symbol;">È </span>B)<br /> </span> </li></ol> <i><br /><span style="font-size:100%;">Given sets A and B, their <b>intersection A </b></span></i> <span style="font-size:100%;"><b> <span style="font-family:Symbol;">Ç</span> <i> B </i> </b> <i> consists of all objects which are in both A and B.</i></span><i><br /></i><br /><span style="font-size:100%;">Thus, A <span style="font-family:Symbol;">Ç</span> B = {x / x <span style="font-family:Symbol;">Î</span> A and x <span style="font-family:Symbol;">Î</span> B} .Two sets A and B are said to be disjoint if and only if A <span style="font-family:Symbol;">Ç</span> B = <span style="font-family:Symbol;">f</span> .</span><br /><br /><span style="font-size:100%;">The intersection operation on sets has the following evident properties:</span><br /><ol type="i"><li> <span style="font-size:100%;">A <span style="font-family:Symbol;font-size:100%;">Ç</span> A = A (Idempotent)<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">Ç</span> B = B <span style="font-family:Symbol;">Ç</span> A (Commutative)<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">Ç</span> <span style="font-family:Symbol;">f</span> = <span style="font-family:Symbol;">f<br /> </span></span></li><li> <span style="font-size:100%;">(A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">Ç</span> C = A <span style="font-family:Symbol;">Ç</span> (B <span style="font-family:Symbol;">Ç</span> C) (Associative)<br /> </span></li><li> <span style="font-size:100%;">A <span style="font-family:Symbol;">Ç</span> B = A if and only if A <span style="font-family:Symbol;">Í</span> B<br /> </span></li><li> <span style="font-size:100%;">(A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">Í</span> A and (A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">Í</span> B<br /> </span></li></ol> <span style="font-size:100%;">An important property connecting <span style="font-family:Symbol;">È</span> and <span style="font-family:Symbol;">Ç</span> is the <b>distributive law</b>.</span><br /><br /><span style="font-size:100%;">(i) A <span style="font-family:Symbol;">Ç</span> (B <span style="font-family:Symbol;">È</span> C) = (A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">È</span> (A <span style="font-family:Symbol;">Ç</span> B)<br />(ii) A <span style="font-family:Symbol;">È</span> (B <span style="font-family:Symbol;">È</span> C) = (A <span style="font-family:Symbol;">È</span> B)<span style="font-family:Symbol;">Ç</span> (A <span style="font-family:Symbol;">È</span> C)</span><br /><b><br /><span style="font-size:100%;">Proof</span></b><span style="font-size:100%;">:</span><br /><br />A <span style="font-family:Symbol;">Ç</span> (B <span style="font-family:Symbol;">È</span> C) = {x / x <span style="font-family:Symbol;">Î</span> A and x <span style="font-family:Symbol;">Î</span> B <span style="font-family:Symbol;">È</span> C}<br /> = {x / x <span style="font-family:Symbol;">Î</span> A and (x <span style="font-family:Symbol;">Î</span> B or x<span style="font-family:Symbol;">Î</span> C)}<br /> = {x / (x <span style="font-family:Symbol;">Î</span> A and x <span style="font-family:Symbol;">Î</span> B) or (x <span style="font-family:Symbol;">Î</span> A and x <span style="font-family:Symbol;">Î</span> C)}<br /> = {x / x <span style="font-family:Symbol;">Î</span> A <span style="font-family:Symbol;">Ç</span> B or x<span style="font-family:Symbol;">Î</span> A <span style="font-family:Symbol;">Ç</span> C}<br /> = (A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">È</span> (A <span style="font-family:Symbol;">È</span> C)<br /><br /><span style="font-size:100%;">Similarly, A <span style="font-family:Symbol;">È</span> (B <span style="font-family:Symbol;">Ç</span> C) = (A <span style="font-family:Symbol;">È</span> B) <span style="font-family:Symbol;">Ç</span> (A <span style="font-family:Symbol;">È</span> C) can be proved.</span><br /><br /><span style="font-size:100%;">It is very interesting to observe that A <span style="font-family:Symbol;">Ç</span> (A <span style="font-family:Symbol;">È</span> B) = A, for A <span style="font-family:Symbol;">Í</span> (A <span style="font-family:Symbol;">È</span> B) and A <span style="font-family:Symbol;">Í</span> A,</span><br /><span style="font-size:100%;">therefore, A <span style="font-family:Symbol;">Ç</span> (A <span style="font-family:Symbol;">È</span> B) = A.</span><br /><br /><span style="font-size:100%;">Similarly, A <span style="font-family:Symbol;">È</span> (A <span style="font-family:Symbol;">Ç</span> B) = A, for (A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">Í</span> A and A <span style="font-family:Symbol;">Í</span> A, hence, A <span style="font-family:Symbol;">È</span> (A <span style="font-family:Symbol;">Ç</span> B) = A.</span><br /><br /><span style="font-size:100%;">Thus, for sets A and B, we have,<br /></span> <b><span style="font-size:100%;"><i>A</i><span style="font-family:Symbol;">Ç</span> <i> (A</i><span style="font-family:Symbol;">È</span> <i> B) = A; A</i><span style="font-family:Symbol;">È</span> <i>(A</i><span style="font-family:Symbol;">Ç</span> <i> B) = A </i>[Absorption law].</span><br /></b><i><br /><span style="font-size:100%;">By the <b>difference, </b>B\A<b>,</b> of the sets B and A, we mean the set of all those objects in B which are not in A.</span></i><br /><br /><span style="font-size:100%;">Thus, B\A = {x : x <span style="font-family:Symbol;">Î </span>B and x <span style="font-family:Symbol;">Ï</span> A}.</span><br /><i><br /><span style="font-size:100%;">The <b>symmetric difference, </b>A<span style="font-family:Symbol;">D</span> B<b>,</b> of sets A and B is the set </span> </i><span style="font-size:100%;">(A\B) <span style="font-family:Symbol;">È</span> (B\A).</span><br /><br /><span style="font-size:100%;"><br /><i>Let X be given set. If we deal with subset of X then we call X is a <b>universal set</b>. If A <span style="font-family:Symbol;">Í</span> X, then the <b>complement</b> </i><sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.1/Image/Image6.gif" width="17" align="absmiddle" height="21" /></sub><i> of A<br /> is defined to be X\A.</i><br />Thus, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit6/Section6.1/Image/Image6.gif" width="17" align="absmiddle" height="21" /> <sub> </sub>= {x / x <span style="font-family:Symbol;">Î</span> X and x <span style="font-family:Symbol;">Ï</span> A}<br /><br />It is clear that whenever we use complements, it is assumed that we are dealing only with subset of some fixed universal set X.</span><br /><br /><span style="font-size:100%;">The following properties can be easily verified:</span><br /><br /><p style="line-height: 150%;" align="left"><b><nobr><br /><a name="6.1.3 Cartesian Products"><span style="font-size:100%;">6.1.3 Cartesian Products</span></a><br /></nobr></b><br /><span style="font-size:100%;">In this section we define Cartesian product of sets. <b><i>Cartesian product</i></b><i> set A </i> <span style="font-family:Symbol;">´</span> <i> B of two arbitrary sets A and B is the set of<br /> all pairs (a, b) such that a<span style="font-family:Symbol;">Î</span> A and b<span style="font-family:Symbol;">Î</span> B.</i> That is, A <span style="font-family:Symbol;">´</span> B = {(a,b) / a <span style="font-family:Symbol;">Î</span> A and b<span style="font-family:Symbol;">Î</span> B}. Note that the sets A and B need not be<br /> distinct in the Cartesian product.</span> <br /><br /><span style="font-size:100%;">In the product A <span style="font-family:Symbol;">´</span> B, the element (a<sub>1</sub>, b<sub>1</sub>) and (a<sub>2</sub>, b<sub>2</sub>) are regarded as equal if and only if a<sub>1</sub> = a<sub>2</sub> and b<sub>1</sub> = b<sub>2</sub>. Thus, if A<br /> consists of m elements a<sub>1</sub>, a<sub>2</sub>, … , a<sub>m</sub> and B consists of n elements b<sub>1</sub>, b<sub>2</sub>, … ,b<sub>n</sub></span> , <span style="font-size:100%;">then A <span style="font-family:Symbol;">´</span> B consists of mn elements (a<sub>i</sub>, b<sub>j</sub>),<br /> 1 <span style="font-family:Symbol;">£</span> i <span style="font-family:Symbol;">£</span> m, 1 <span style="font-family:Symbol;">£</span> j <span style="font-family:Symbol;">£</span> n.</span> <b><nobr><br /><a name="6.1.4 Motivation of Boolean Algebra"><span style="font-size:100%;">6.1.4 Motivation of Boolean Algebra</span></a><br /><br /></nobr></b><span style="font-size:100%;">In this section we shall discuss the motivation of Boolean algebra. From the Section 6.1.2, Algebra of Sets, it is interesting to<br /> observe that, if A is any set, then the binary operations, <span style="font-family:Symbol;">Ç</span> and <span style="font-family:Symbol;">È</span> on the set <i>P</i>(A) satisfy the following properties</span><br /></p><ol type="i"><li> <span style="font-size:100%;">Commutative law,<br /> </span></li><li> <span style="font-size:100%;">Associative law,<br /> </span></li><li> <span style="font-size:100%;">Absorption law,<br /> </span></li><li> <span style="font-size:100%;">Idempotent property and<br /> </span></li><li> <span style="font-size:100%;">Distributive law.</span></li></ol><br /><span style="font-size:100%;">and the uninary " - " operation on <i>P</i>(A) satisfies the De Morgan's law. So it is tempting to ask the question:<br /><b>"Are there other sets like <i>P</i>(A) and binary operations like <span style="font-family:Symbol;">È</span> and <span style="font-family:Symbol;">Ç</span> satisfying commutative law, associative law,<br />absorption law, idempotent property and distributive law and uninary operation "</b> <b> -</b> <b> "</b></span><b> <span style="font-size:100%;">satisfying De Morgan's law?".</span><br /></b><br /><span style="font-size:100%;">Indeed there are many such sets and such binary and uninary operations on such sets satisfying the above mentioned laws and<br />property. <i> <b> In fact, one of the main motivations of Boolean Algebra is a search of such sets and understanding their<br /> algebraic structure.</b></i></span>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-90691318797482156452008-08-12T04:29:00.001-07:002008-12-23T01:55:02.554-08:00Computable and Non-computable Functions<b><a name="Computable and Non-computable Functions"></a></b><br /> <br />Turing machine may be viewed as a computing machine that computes the functions which are defined from integers into integers.<br /> The traditional approach is to represent integers as unary; the integer i <span style="font-family:Symbol;">³</span> 0 is represented by the string 0<sup>i</sup>. If a function has k<br /> arguments, i<sub>1</sub>, i<sub>2</sub>,…, i<sub>k</sub>, then these integers are initially placed on the tape separated by 1's as .[Different authors<br /> have different approach on the representation of inputs, generally, representation of inputs is not unique].<br /><br />On the input , if the TM halts with a tape consisting of 0<sup>m</sup> for some m, after the computation on this input, then<br /> we say that f(i<sub>1</sub>,i<sub>2</sub>, … , i<sub>k</sub>) = m, where f is the function of k arguments computed by this Turing machine.<br /><i><br />A function f with integer arguments is said to be <b>computable</b> if there exists a Turing machine M such that M would halt<br /> on every input of f.<br /></i> <br /> <b> <a name="5.2.1 Example of Computable Functions">5.2.1 Example of Computable Functions<br /> </a> </b> <br />In this subsection we construct or design a Turing machine TM that will compute the "<i>proper subtraction function</i>". That is, we<br /> show that the proper subtraction function is computable. Similarly one can design Turing machine for all the recursive functions.<b> <br /> Thus, it can be proved that all recursive functions are computable functions.</b><br /> <br /> <b> <a name="5.3.1.1 Proper Subtraction Function is a Computable Function">5.2.1.1 Proper Subtraction Function is a Computable Function<br /> </a> </b> <br />Recall that proper subtraction f(m,n) is defined to be m - n for m > n and zero for m <span style="font-family:Symbol;">£</span> n. Now we design a Turing machine,<br /> which will compute proper subtraction for given two integers.<br /><br />The Turing machine TM, M is defined as follows:<br /><br />M = ({q<sub>0</sub>, q<sub>1</sub>, … , q<sub>6</sub>}, {0,1},{0,1,B}, <span style="font-family:Symbol;">d</span>, q<sub>0</sub>, B, <span style="font-family:Symbol;">f</span>), where <span style="font-family:Symbol;">d</span> is defined below. The TM starts with 0<sup>m</sup>10<sup>n</sup> on its tape halts<br /> with 0<span style="font-size:100%;"><sup>m <span style="font-family:Symbol;">-</span> n</sup></span> on its tape. M repeatedly replaces its leading 0 by blank, then searches right for a 1 followed by a 0 and changes the<br /> 0 to 1. Next, M moves left until it encounters a blank and then repeats the cycle. The repetition ends if<br /><ol type="i"><li> Searching right for a 0, M encounters a blank. Then, the n0's in 0<sup>m</sup>10<sup>n</sup> have all been changed to 1’s, and n + 1 of the m 0’s have been changed to B. M replaces the n+1 1’s by a 0 and n B's leaving m - n 0's on its tape.<br /> </li><li> Beginning the cycle, M cannot find a 0 to change to a blank, because the first m 0's already have been changed. Then n <span style="font-family:Symbol;">³</span> m, so m <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/dotline.gif" width="20" align="absmiddle" border="0" height="20" /> n = 0, M replaces all remaining 1’s and 0’s by B.</li></ol><br />The transition function <span style="font-family:Symbol;">d</span> is described below.<br /><ol><li> <span style="font-family:Symbol;">d</span>(q<sub><span style="font-size:85%;">0</span></sub>, 0) = (q<sub>1</sub>, B, R)<br /> Begin the cycle, replace the leading 0 by B.<br /> </li><li> <span style="font-family:Symbol;">d</span>(q<sub>1</sub>, 0) = (q<sub>1</sub>, 0, R)<span style="font-family:Symbol;"><br /> d</span>(q<sub>1</sub>, 1) = (q<sub>2</sub>, 1, R)<br /> Searching right, looking for the first 1.<br /> </li><li> <span style="font-family:Symbol;">d</span>(q<sub>2</sub>, 1) = (q<sub>2</sub>, 1, R)<br /> <span style="font-family:Symbol;">d</span>(q<sub>2</sub>, 0) = (q<sub>3</sub>, 1, L)<br /> Searching right past 1’s until encountering a 0. Change that 0 to 1.<br /> </li><li> <span style="font-family:Symbol;">d</span>(q<sub>3</sub>, 0) = (q<sub>3</sub>, 0, L)<br /> <span style="font-family:Symbol;">d</span>(q<sub>3</sub>, 1) = (q<sub>3</sub>, 1, L)<br /> <span style="font-family:Symbol;">d</span>(q<sub>3</sub>, B) = (q<sub><span style="font-size:85%;">0</span></sub>, B, R)<br /> Move left to a blank. Enter state q<sub><span style="font-size:85%;">0</span></sub> to repeat the cycle.<br /> </li><li> <span style="font-family:Symbol;">d</span>(q<sub>2</sub>, B) = (q<sub>4</sub>, B, L)<br /> <span style="font-family:Symbol;">d</span>(q<sub>4</sub>, 1) = (q<sub>4</sub>, B, L)<br /> <span style="font-family:Symbol;">d</span>(q<sub>4</sub>, 0) = (q<sub>4</sub>, 0, L)<br /> <span style="font-family:Symbol;">d</span>(q<sub>4</sub>, B) = (q<sub>6</sub>, 0, R)<br /> If in state q<sub>2</sub> a B is encountered before a 0, we have situation (i) described above. Enter state q<sub>4</sub> and move left, changing<br /> all 1's to B's until encountering a B. This B is changed back to a 0, state q<sub>6</sub> is entered, and M halts.<br /> </li><li> <span style="font-family:Symbol;">d</span>(q<sub><span style="font-size:85%;">0</span></sub>, 1) = (q<sub>5</sub>, B, R)<br /> <span style="font-family:Symbol;">d</span>(q<sub>5</sub>, 0) = (q<sub>5</sub>, B, R)<br /> <span style="font-family:Symbol;">d</span>(q<sub>5</sub>, 1) = (q<sub>5</sub>, B, R)<br /> <span style="font-family:Symbol;">d</span>(q<sub>5</sub>, B) = (q<sub>6</sub>, B, R)<br /> If in state q<sub><span style="font-size:85%;">0</span></sub> a 1 is encountered instead of a 0, the first block of 0's has been exhausted, as in situation ( ii ) above. M enters<br /> state q<sub>5</sub> to erase the rest of the tape, then enters q<sub>6</sub> and halts.</li></ol> <b><br />An Illustration of computation of M is a given below:<br /><br />On the input 0010:<br /><br /><br /><br />On the input 0100:<br /><br /><br /></b><p style="line-height: 150%;" align="left"><b><nobr><br /><a name="5.2.2 Computability and Non-computability">5.2.2 Computability and Non-computability</a><br /> <br /></nobr></b><i>A function or a (decision type) problem is said to be computable, if there exist a Turing machine which computes the<br /> function or answer the problem. [i.e., the TM will halt on all the inputs and gives the correct output for all the input].<br /> Otherwise, the function is said to be <b>non-computable.<br /></b></i><br />Note that there exist Turing Machines to compute the initial functions<br /></p><ol type="i"><ol type="i"><li> Z (x) = 0, for all x <span style="font-family:Symbol;">Î</span> N (the zero function)<br /> </li><li> S (x) = x+1, for all x <span style="font-family:Symbol;">Î</span> N (the successor function)<br /> </li><li> U<sub>i</sub><sup>n</sup>(x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, …, x<sub>i</sub>, ..., x<sub>n</sub>) = x<sub>i </sub>(the projection function)</li></ol></ol><br />There also exist Turing machines to compute composition, recursion and minimization operations. <b>As every partial function is<br /> obtained from the initial functions by a finite number of applications of the operations of composition, recursion and minimization, thus, there exist Turing machine that can compute every partial recursive function. Thus, every partial recursive function is computable. On the other hand it can be proved that every Turing computable function is partial recursive. </b> <br /><br />Here we give <b>some examples of non-computable problems.<br /></b><br />"Halting problem" for Turing machine is one of the popular examples for non-computability of Turing machines. This is the problem<br />of finding a decision procedure which would enable one to determine in a finite number of steps, given any Turing machine T and<br /> premarked tape 't', whether or not the machine will ever make <span style="font-family:Symbol;">d</span>(s<sup>j</sup>, a<sup>j</sup>) = HALT. Equivalently the problem is to construct a Turing<br /> machine, which will accept just those pairs (T, t) that will come to HALT, and reject just those pairs (T, t) that will not come to<br /> HALT. It can be proved theoretically that there cannot exist any Turing machine, which will perform this job. Thus, the <b><br />"HALTING PROBLEM IS NONCOMPUTABLE".<br /></b><br />Other interesting example of noncomputable functions are the following.<br /><br />For any programming language, to determine whether or not<br /><br />(a) a given program that can loop for ever on some input,<br /><br />(b) a given program that can ever produce an output on given input,<br /><br />(c) a given program that eventually halts on the given input.<br /><b><br />Note to the Reader<br /></b><br />To understand and appreciate the depth of the concepts of computability and non-computability, one has to have good<br /> understanding in formal languages and theory of computing, so readers are advised to refer good books for detailed study on<br /> these topics [some of the books are given in the <b>Reference</b>]. Here we have introduced this topic mainly because; "all partial<br /> recursive functions are computable. Also understanding about the Turing machine will help to study other important Theoretical<br /> Computer Sciences topics like Computation and Complexity etc.Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-83984715548963631402008-08-12T04:28:00.001-07:002008-12-23T01:55:02.554-08:00Turing MachineIn this Chapter, we briefly introduce computable and non-computable functions. For the detailed understanding about this topic<br /> one can refer the books [1,2,3,4].<br /><br />In the mid of 1930’s, mathematicians and logicians were rigorously trying to define computability and algorithm. In 1934,<br /> Kurt Gödel pointed out that primitive recursive functions can be computed by a finite procedure [that is, an algorithm]. He also hypothesized that any function computable by a finite procedure can be specified by a recursive function. Around 1936, Turing and Church independently designed a "Computing machine" (Later termed as Turing machine) which can carry out a finite procedure.<b><span style="font-size:130%;"><a name="5.1 Turing Machine"><br /><br /> Turing Machine<br /><br /></a></span></b>In this section we define Turing machines. The basic model of a Turing machine illustrated in the following Figure.1 has a finite<br /> control, an input tape that is divided into cells, a tape head that scans one cell on the tape at a time. The tape has a leftmost cell<br /> but is infinite to the right.<br /> <br /><br /><br /><br /><br />Each cell of the tape may hold exactly one of a finite number of tape symbols. Initially the n leftmost cells, for some finite n <span style="font-family:Symbol;">³</span> 0,<br /> hold the input, which is a string of symbols chosen from a subset of the tape symbols called input symbols. The remaining infinity<br /> of cells each hold the blank, which is a special tape symbol that is not an input symbol.<br /><br />In one move the Turing machine, depending upon the symbol scanned by the tape head and the state of the finite control, <ol><li> changes state,</li><li> prints a symbol on the tape cell scanned, replacing what was written there, and</li><li> moves its head left or right one cell.</li></ol> Formally, a Turing machine (TM) is denoted by<br /><br />M = (Q, <span style="font-family:Symbol;">S</span> , <span style="font-family:Symbol;">G</span> , <span style="font-family:Symbol;">d</span> , q<sub><span style="font-size:85%;">0</span></sub>, B, F), where<br />Q is the finite set of <i>states</i>,<span style="font-family:Symbol;"><br />G</span> is the finite set of allowable <i>tape symbols</i>,<br />B, a symbol of <span style="font-family:Symbol;">G</span> , is called <i>blank</i>,<span style="font-family:Symbol;"><br />S</span> , a subset of <span style="font-family:Symbol;">G</span> not including B, is the set of <i>input symbols</i>,<span style="font-family:Symbol;"><br />d</span> is the <i>next move function</i>, a mapping from Q <span style="font-family:Symbol;">´</span> <span style="font-family:Symbol;">G</span> to Q <span style="font-family:Symbol;">´</span> <span style="font-family:Symbol;">G</span> <span style="font-family:Symbol;">´</span> {L, R} (<span style="font-family:Symbol;">d</span> may however be undefined for some arguments),<br />q<sub><span style="font-size:85%;">0</span></sub> in Q is the initial<i> state</i>,<br />F <span style="font-family:Symbol;">Í</span> Q is the set of <i>final states</i>.<i><br /><br />The functional status of a Turing machines TM at a given instance can be defined and termed as "<b>Instantaneous<br /> description</b>" (ID) of the Turing machine M by <span style="font-family:Symbol;">a</span><sub>1</sub>q<span style="font-family:Symbol;">a</span><sub>2</sub>. Here, q, the current state of M, is in Q; <span style="font-family:Symbol;">a</span><sub>1</sub><span style="font-family:Symbol;">a</span> <sub>2</sub> is the string in <span style="font-family:Symbol;">G</span> <span style="font-size:100%;"><sup>*</sup></span>,<br />that is the contents of the tape upto the rightmost nonblank symbol or the symbol to the left of the head, whichever is<br /> the rightmost</i>. (Observe that the blank B may occur in <span style="font-family:Symbol;">a</span><sub>1</sub><span style="font-family:Symbol;">a</span><sub>2</sub>).<br /><br />We assume that Q and <span style="font-family:Symbol;">G</span> are disjoint to avoid confusion. The tape head is assumed to be scanning the leftmost symbol of <span style="font-family:Symbol;">a</span><sub>2</sub> or<br /> if <span style="font-family:Symbol;">a</span><sub>2</sub> = <span style="font-family:Symbol;">L</span> (the empty string), then the head is scanning a blank.<br /><br />We define a <b><i>move of a</i> <i>Turing machine</i></b> (TM) M as follows:<br />Let X<sub>1</sub>X<sub>2 </sub>… X<sub>i-1</sub> q X<sub>i </sub>… X<sub>n</sub> be an ID.<br />Suppose <span style="font-family:Symbol;">d</span> (q , X<sub>i</sub>) = (p, Y, L), where if i – l = n, then X<sub>i</sub> is taken to be B. If i = l, then there is no next ID, as the tape head is<br /> not allowed to fall off the left end of the tape. If i > 1, then we write<br />X<sub>1</sub>X<sub>2 </sub>… X<sub>i-1</sub> q X<sub>i </sub>… X<sub>n</sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/M_under.gif" width="22" align="absmiddle" border="0" height="22" /> X<sub>1</sub>X<sub>2 </sub>… X<sub>i-2</sub> p X<sub>i-1</sub>Y X<sub>i+1</sub> … X<sub>n</sub> …….. (1)<br /><br />However, if any suffix of X<sub>i-1</sub>Y X<sub>i+1</sub>… X<sub>n</sub> is completely blank, that suffix is deleted in (1). Alternatively, suppose <span style="font-family:Symbol;"><br />d</span> (q, X<sub>i </sub>) = (p, Y, R), then we write,<br />X<sub>1</sub>X<sub>2 </sub>… X<sub>i-1</sub> q X<sub>i</sub> X<sub>i+1</sub>… X<sub>n</sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/arrow.gif" width="23" align="absmiddle" border="0" height="26" /> X<sub>1</sub> X<sub>2</sub> … X<sub>i-1</sub> Y p X<sub>i+1</sub> … X<sub>n </sub> <sub> </sub>……… (2)<br /><br />Note that in the case i-1 = n, the string X<sub>i</sub> X<sub>i+1</sub>… X<sub>n</sub> is empty, and the right side of (2) is larger than the left side.<br /><br />If one ID results from another by some finite number of moves, including zero moves, then we say the ID's are related and one ID<br /> yields the other ID.<br /><br />We give the definition of <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Image174.gif" width="13" height="20" /></sub> in the form of a table called the <i> transition table</i>.<br />If <span style="font-family:Symbol;">d</span>(q, a) = (<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Image175.gif" width="45" align="absmiddle" height="24" /></sub>), we write <span style="font-family:Symbol;">abg</span> under "a-column" and "q-row". So if we get <span style="font-family:Symbol;">abg</span> in the table, it means that <span style="font-family:Symbol;">a</span> is written in<br /> the current cell, <span style="font-family:Symbol;">b</span> gives the movement of the head L or R and <span style="font-family:Symbol;">g</span> denotes the new state into which the Turing machine enters.<br /><br />Consider, for example, a Turing machine with five states q<sub>1</sub>, …, q<sub>5</sub>, where q<sub>1</sub> is the initial state and q<sub>5</sub> is the (only) final state.<br /> The tape symbols are 0,1 and B. The transition table given in the following Table 1 describes <span style="font-family:Symbol;">d</span>.<b><br /><br /> </b> <b>Table 1</b>: <b>Transition Table of a Turing Machine<br /></b> <center> <table width="541" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td rowspan="2" valign="top" width="25%" height="27"><b> </b><p style="line-height: 150%;" align="center"><b>Present state</b></p></td> <td colspan="3" valign="top" width="75%" height="27"> <p style="line-height: 150%;" align="center"><b>Tape symbols</b></p></td> </tr> <tr> <td valign="top" width="25%" height="7"><b> </b><p style="line-height: 150%;" align="center"><b>B</b></p></td> <td valign="top" width="25%" height="7"><b> </b><p style="line-height: 150%;" align="center"><b>0</b></p></td> <td valign="top" width="25%" height="7"><b> </b><p style="line-height: 150%;" align="center"><b>1</b></p></td> </tr> <tr> <td valign="top" width="25%" height="30"> <p style="line-height: 150%;" align="center"><span style="font-family:Symbol;">®</span> q<sub>1</sub></p></td> <td valign="top" width="25%" height="30"> <p style="line-height: 150%;" align="center">1Lq<sub>2</sub></p></td> <td valign="top" width="25%" height="30"> <p style="line-height: 150%;" align="center">0Rq<sub>1</sub></p></td> <td valign="top" width="25%" height="30"> <p style="line-height: 150%;" align="center">__</p></td> </tr> <tr> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">q<sub>2</sub></p></td> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">BRr<sub>3</sub></p></td> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">0Lq<sub>2</sub></p></td> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">1Lq<sub>2</sub></p></td> </tr> <tr> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">q<sub>3</sub></p></td> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">__</p></td> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">BRq<sub>4</sub></p></td> <td valign="top" width="25%" height="27"> <p style="line-height: 150%;" align="center">BRq<sub>5</sub></p></td> </tr> <tr> <td valign="top" width="25%" height="25"> <p style="line-height: 150%;" align="center">q<sub>4</sub></p></td> <td valign="top" width="25%" height="25"> <p style="line-height: 150%;" align="center">0Rq<sub>5</sub></p></td> <td valign="top" width="25%" height="25"> <p style="line-height: 150%;" align="center">0Rq<sub>4</sub></p></td> <td valign="top" width="25%" height="25"> <p style="line-height: 150%;" align="center">1Rq<sub>4</sub></p></td> </tr> <tr> <td valign="top" width="25%" height="43"> <p style="line-height: 150%;"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio20.gif" width="45" border="0" height="41" /></p> </td> <td valign="top" width="25%" height="43"> <p style="line-height: 150%;" align="center">0Lq<sub>2</sub></p></td> <td valign="top" width="25%" height="43"> <p style="line-height: 150%;" align="center">__</p></td> <td valign="top" width="25%" height="43"> <p style="line-height: 150%;" align="center">__</p></td> </tr> </tbody></table> </center><br />The initial state is marked with <span style="font-family:Symbol;">®</span> and final state with <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio21.gif" width="22" border="0" height="21" /> .<br /><br />Computation of a Turing machine M is a sequence of ID’s such that an ID in a sequence yields its next ID under the definition<br /> of transition function <span style="font-family:Symbol;">d</span> .<br /><br />We give below the computation of the above TM on the input string 00B.<br /><br />Q100B <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> 0q<sub>1</sub>0B <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> 00q<sub>1</sub>B <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> 0q<sub>2</sub>01 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> q<sub>2</sub>001 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> q<sub>2</sub>B001 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> Bq<sub>3</sub>001 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBq<sub>4</sub>01 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BB0q<sub>4</sub>1 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BB1q<sub>4</sub>B <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> <br />BB010q<sub>5</sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BB01q<sub>2</sub>00 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BB0q<sub>2</sub>100 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBq<sub>2</sub>0100 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> Bq<sub>2</sub>Bo100 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBq<sub>3</sub>0100 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBBq<sub>4</sub>100 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBB1q<sub>4</sub>00 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> <br />BBB10q<sub>4</sub> 0 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBB100q<sub>4</sub>B<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBB1000q<sub>5</sub>B <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" />BBB100q<sub>2</sub>00 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBB10q<sub>2</sub>000 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBB1q<sub>2</sub>0000 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" />BBBq<sub>2</sub>10000 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> <br /> BBq<sub>2</sub>B10000 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBBq<sub>3</sub>1000 <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit5/Section5.1/Image/Sectio1.gif" width="21" border="0" height="22" /> BBBBq<sub>5</sub>0000Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-58474185585734311742008-08-12T04:27:00.000-07:002008-12-23T01:55:02.554-08:00Recursive Functions<b><a name="4.1.1 Basics">Basics</a></b>
<br /><i>
<br />Consider only those functions whose arguments and values are natural numbers. Such functions are called <b>
<br />number - theoretic</b></i>. In general number - theoretic function in
<br />
<br />n variable is considered as f <x<sub>1</sub>, x<sub>2,</sub>...,x<sub>n</sub>>.
<br /><i>
<br />Any function f : N<sup>n</sup> <span style="font-family:Symbol;">®</span> N is called <b>total</b> because it is defined for every n-tuple in N<sup>n</sup>.
<br />
<br />On the other hand, if f: D <span style="font-family:Symbol;">®</span> N where D <span style="font-family:Symbol;">Í</span> N<sup>n</sup>, then f is called <b>partial.
<br /></b></i><b>
<br />
<br />
<br />Examples:
<br /></b> <ol><li> f<x,> = x+y which is defined for all x, y<span style="font-family:Symbol;"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image18.gif" width="13" height="13" /></span> N and hence is a total function.</li><li> g<x,> = x-y which is defined for only those x, y <span style="font-family:Symbol;"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image18.gif" width="13" height="13" /> </span>N, which satisfy x > y.
<br />Hence g <x,> is partial. </li></ol> Every total function of n variables is also a n - ary operation on N.
<br />
<br />
<br />
<br /><b>Initial Functions:</b>
<br />
<br />Consider a set of three functions called the initial functions, which are used in defining other functions by induction.
<br />
<br />Z : Z(x) = 0 Zero function
<br />
<br />S : S(x) = x+1 Successor function
<br />
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio13.gif" width="337" border="0" height="37" />
<br />
<br />The projection function is also called generalized identity function.
<br /><b>
<br />Examples:</b>
<br />
<br /><span style="font-size: 12pt; font-family: Times New Roman;"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio14.gif" width="103" border="0" height="27" /> </span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio15.gif" width="137" border="0" height="31" />
<br />
<br />The operation of composition will be used to generate other functions. Composition of functions idea can be extended to
<br /> functions of more than one variable.
<br />
<br />For example, let f<sub>1</sub><x,> , f<sub>2</sub><x,> and g<x,> be any three functions. The composition of g with f<sub>1</sub>, and f<sub>2</sub> is a function h
<br /> given by h<x,> = g <f<sub>1</sub><x,>, f<sub>2</sub><x,>>
<br />
<br />
<br />For h to be non-empty, it is necessary that the domain of g include <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image134.gif" width="76" align="absmiddle" height="32" /> where <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image135.gif" width="29" align="absmiddle" height="32" />and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image136.gif" width="30" align="absmiddle" height="32" />are the ranges of f<sub>1</sub> and f<sub>2</sub>
<br /> respectively. Also the domain of h is <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image137.gif" width="80" align="absmiddle" height="32" /> where <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image139.gif" width="29" align="absbottom" height="32" />and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image140.gif" width="30" align="absbottom" height="32" />are the domains of f<sub>1</sub> and f<sub>2</sub> respectively.
<br />
<br />If f<sub>1</sub>, f<sub>2 </sub>and g are total, then h is also total. In general, let f<sub>1</sub>, f<sub>2</sub>, …, f<sub>n</sub> each be partial functions of m variables, and let g be a
<br /> partial function of n variables. Then the composition of g with f<sub>1</sub>,f<sub>2</sub>,…,f<sub>n</sub> produces a partial function h given by
<br />
<br />h<x<sub>1</sub>,x<sub>2</sub>,....,x<sub>m</sub>> = g<f<sub>1</sub><x<sub>1</sub>,x<sub>2</sub>,…,x<sub>m</sub>>,…,f<sub>n</sub><x<sub>1</sub>,x<sub>2</sub>,…,x<sub>m</sub>>>.
<br />
<br />It is assumed that the domain of g includes the n - tuples <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image141.gif" width="77" align="absmiddle" height="32" />, I<sub>n</sub>={1,2,.....,n} and <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image142.gif" width="28" align="absmiddle" height="32" /></sub>denotes the range of f<span style="font-size:100%;"><sub>i</sub></span>. Also the
<br /> domain of h is given by <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image143.gif" width="56" align="absbottom" height="37" />,where <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image144.gif" width="28" align="absbottom" height="32" />is the domain of f<span style="font-size:100%;"><sub>i</sub></span><span style="font-size:130%;">. </span>The function h is total iff f<sub>1</sub>, f<sub>2</sub>,...,f<sub>n</sub>, and g are total.
<br />
<br />Let f<sub>1</sub><x,> = x+y,
<br />
<br />f<sub>2</sub><x,y> = xy+y<sup>2</sup> and
<br />
<br />g<x,y> = xy.
<br />
<br />Then, h<x,y> = g<f<sub>1</sub><x,y>,f<sub>2</sub><x,y>>
<br /> = g<x+y,xy+y<sup>2</sup>>
<br /> = (x+ y)(xy+y<sup>2</sup>).
<br />
<br />Here h is total, because f<sub>1</sub>, f<sub>2</sub>, and g are all total.
<br />
<br />Given a function f<x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>> of n variables and consider n - 1 of the variables as fixed and vary only the remaining variable
<br /> over the set of natural numbers or over a subset of it. For example, fix x and vary y in f<x,y> to obtain
<br /> f<x,0>,f<x,1>,f<x,2>,……
<br /><b>
<br /> Example:
<br /></b>
<br />If f<x,y> = x+y and f<2,0> = 2, find f<2,3>.
<br /><b>
<br />Solution:
<br /></b>
<br />First evaluate f<2,1>, f<2,2> and finally f<2,3>. Each functional value is computed by adding 1 to the previous value until the
<br /> desired result is obtained.
<br />
<br />The computation of f<2,3> is
<br />
<br />f<2,3> = [(f<2,0>+1)+1]+1
<br /> = [(2+1)+1)+1
<br /> = [3+1]+1
<br /> = 4+1
<br /> = 5.
<br />
<br /><b>Recursion:</b>
<br />
<br />It is assumed that we have a mechanism by which we can determine the value of the function when an argument is zero and also
<br /> its value for the argument n + 1 from the value of the function when the argument is n. <i>The arguments which are considered to
<br /> be fixed are called <b>parameters</b>, while the one which is assumed to vary is considered a <b>variable</b></i>.
<br /><i>
<br />The following operation which defines a function </i> f<x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>,y> <i> of n+1 variables by using two other functions
<br /></i> g<x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>> <i> and </i> h<x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>,y,z> <i> of n and n+2 variables, respectively, is called <b>recursion</b>.
<br />
<br /></i> f<x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>,0> = g<x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>>
<br /> f<x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>,y+1> = h<x<sub>1</sub>, x<sub>2</sub>,...,x<sub>n </sub>,y,f<x<sub>1</sub>,x<sub>2</sub>,...,x<sub>n</sub>,y>>
<br />
<br />
<br />In this definition, the variable y is assumed to be the inductive variable in the sense that the value of f at y +1 is expressed in terms
<br /> of the value of f at y. The variables x<sub>1</sub>,x<sub>2</sub>,…, x<sub>n </sub>are treated as parameters and are assumed to remain fixed throughout the
<br />definition. Also it is assumed that both the functions g and h are known. We shall now impose restrictions on g and h which will
<br /> guarantee that the function f which is defined recursively, as above, can actually be computed and is total.
<br /><i>
<br />
<br /></i> <p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Section4.1.htm#4.1%20%20%20%20Recursive%20Functions">Back to top</a></span> </p><p style="line-height: 150%;" align="left"><i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio2.gif" width="767" border="0" height="10" />
<br /></i> <nobr>
<br /><a name="4.1.2 Primitive Recursive Function"><b>4.1.2 Primitive Recursive Function</b></a>
<br /><i>
<br />A function f is called<b> primitive recursive</b> iff it can be obtained from the initial functions by a finite number of operations
<br /> of composition and recursion
<br /></i>
<br />From the definition it follows that it is not always necessary to use only the initial functions in the construction of a particular
<br />primitive recursive function. If we have a set of functions f<sub>1</sub>,f<sub>2</sub>,…,f<sub>k</sub> which are primitive recursive, then we can use any of these
<br />functions along with the initial functions to obtain another primitive recursive function, provided we restrict ourselves to the
<br /> operations of composition and recursion only.
<br />
<br />In the examples given here, first we construct some primitive recursive functions by using the initial functions alone, and then we
<br /> use these functions wherever required in order to construct other primitive recursive functions. All primitive recursive functions
<br /> are total.
<br /><b>
<br />Example 1:</b>
<br />
<br />Show that the function f<x,y> = x+y is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>x+(y+1) = (x+y)+1.
<br />Therefore f<x,y+1> = f<x,y>+1 = S(f<x,y>) and
<br />
<br />f<x,0> = x.
<br />
<br />Define f<x,y> as
<br />
<br />f<x,> = x = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image145.gif" width="42" align="absmiddle" height="26" /> and
<br />f<x,> = S (<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image146.gif" width="32" align="absmiddle" height="28" /><> > ).
<br />
<br />Here the basic function is
<br /> g(x) = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image147.gif" width="42" align="absbottom" height="26" />.
<br />and the inductive step function is
<br /> h<x,> = S (<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image148.gif" width="32" align="absmiddle" height="28" /><x,>).
<br />For example using these definition, we can compute the value of f<2,> we have
<br />f<2,> = 2.
<br />
<br />f<2,> = S(f<2,>
<br /> = S(S(f<2,>))
<br /> = S(S(S(f<2,>)))
<br /> = S(S(S(S(f<2,>))))
<br /> = S(S(S(S(2))))
<br /> = S(S(S((3))))
<br /> = S(S(4)) = S (5) = 6
<br /><b>
<br />Example 2:</b>
<br />
<br />Using recursion, define the multiplication function * given by g<x,> = x <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image149.gif" width="12" height="13" /> y.
<br /><b>
<br />Solution:
<br /></b>
<br />Since g<x,> = 0 and g<x,> = g<x,> + x, we write
<br />g<x,> = Z(x).
<br />g(x, y + 1) = f < <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image150.gif" width="34" align="absmiddle" height="28" /><>>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image151.gif" width="38" align="absmiddle" height="28" /><x,>>> where f is the addition function given in Example 1.
<br />
<br />
<br /><b>The following are some of the primitive recursive functions which are used frequently.</b>
<br /><ol><li> Sign function or non zero test function, sg :</li> sg(0) = 0 sg (y+1) = 1 or
<br /> sg(0) = Z(0) sg(y+1) = S(Z(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image152.gif" width="34" align="absbottom" height="28" /><>)).
<br /><li> Zero test function, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image153.gif" width="18" align="absmiddle" height="25" /> : <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image153.gif" width="18" align="absbottom" height="25" />(0) = 1, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image153.gif" width="18" align="absbottom" height="25" />(y + 1) = 0.
<br /> </li><li> Predecessor function, P :
<br /> P(0) = 0 P(y+1) = y = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image154.gif" width="34" align="absmiddle" height="28" />(y, P(y)).
<br /> Note that
<br /> P(0) = 0 , P(1) = 0 , P(2) = 1 , P(3) = 2.
<br /> </li><li> Odd and even parity function, P<span style="font-size:100%;">r</span> :
<br /> P<span style="font-size:100%;">r</span>(0) = 0, P<span style="font-size:100%;">r</span>(y + 1) = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image155.gif" width="18" align="absmiddle" height="24" />(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image152.gif" width="34" align="absmiddle" height="28" /><y, size="3">r</span> (y)>))
<br /> P<span style="font-size:100%;"><sub>r</sub></span>(0) = 0, P<span style="font-size:100%;"><sub>r</sub></span>(1) = 1, P<span style="font-size:100%;"><sub>r</sub></span>(2) = 0, P<span style="font-size:100%;"><sub>r</sub></span>(3) = 1, ……
<br /> </li><li> Proper subtraction function, <u> • </u> :
<br /> x <u> • </u> 0 = x, x <u> • </u> (y + 1) = p ( x<u> • </u> y).
<br /> Note that x <u> • </u> y = 0 for x<y> • </u> y = x - y for x <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image27.gif" width="13" height="16" /> y.
<br /> </li><li> Absolute value function, | | :
<br /> |x – y| = (x <u> • </u> y) + (y <u> • </u> x)
<br /> </li><li> min (x , y) = minimum of x and y
<br /> min (x , y) = x <u> • </u> (x <u> • </u> y).
<br /> max <x,> = maximum of x and y
<br /> = y + (x <u> • </u> y).
<br /> </li><li> The square function, f(y) = y<sup>2</sup>
<br /> f(y) = y<sup>2 </sup>= <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image156.gif" width="29" align="absmiddle" height="28" />(y) * <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image157.gif" width="29" align="absmiddle" height="28" />(y) </li></ol>
<br /><b>
<br />
<br />Example 3:</b>
<br />
<br />Show that f<x,> = x<span style="font-size:100%;"><sup>y</sup></span> is primitive recursive function.
<br /><b>
<br />Solution:
<br /></b>
<br />Note that x<sup>0</sup> = 1 for x <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image2.gif" width="14" height="14" /> 0 and we put x<sup>0</sup> = 0 for x = 0.
<br />
<br />Also x<span style="font-size:100%;"><sup>y+1</sup></span><span style="font-size:130%;"> </span>= x<span style="font-size:100%;"><sup>y</sup></span><span style="font-size:130%;"><sup> </sup> </span>* x<sup><span style="font-size:130%;"> </span></sup>; hence f<> = x<span style="font-size:100%;"><sup>y</sup></span> is defined as
<br />
<br />f<x,> = sg(x).
<br />
<br />f<x,> = x * f<x,>
<br /> = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image158.gif" width="34" align="absmiddle" height="28" /><>> * <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image159.gif" width="34" align="absmiddle" height="28" /><>>
<br />
<br />
<br /><b>
<br />Example 4:</b>
<br />
<br />Show that if f<x,> defines the remainder upon division of y by x, then it is a primitive recursive function.
<br /><b>
<br />Solution:
<br /></b>
<br /> For y=0, f<x> = 0.
<br />Also the value of f<x,> increases by 1 when y is increased by 1, until the value becomes equal to x, in which case it is put
<br /> equal to 0 and the process continues.
<br />
<br />We therefore build a function which increases by 1 each time y increases by 1, that is, S(f<x,>).
<br />
<br />Now we multiply this function by another primitive recursive function which becomes 0 whenever
<br />S(f<x,>) = x .
<br />
<br />Also this other function must be 1 whenever S(f<x,>) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image2.gif" width="14" height="14" />x.
<br />But S(f<x,>) is always <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image19.gif" width="13" height="16" />x and hence such a function is sg ( x <u> • </u> S(f<x,>)).
<br />
<br />Thus the required definition of f<x,>is
<br />f<x,> = 0
<br />
<br />f<x,> = S(f<x,>) * sg (x <u> • </u> S(f<x,>)).
<br /><b>
<br />
<br />
<br />Example 5:</b>
<br />
<br />Show that the function [x/ 2] which is equal to the greatest integer which is <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image19.gif" width="13" height="16" /> x/2 primitive recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />Now [0/2] = 0.
<br />[1/2] = 0.
<br />[2/2] = 1.
<br />[3/2] = 1, etc., so that [x/2] = x/2 when x is even and [x/2] = (x-1)/2 when x is odd.
<br />
<br />In order to distinguish between even and odd functions, we have defined the parity function, which is primitive recursive.
<br />[0/2] = 0,
<br />
<br />[(y+1)/2] = [y/2] + P<span style="font-size:100%;"><sub>r</sub></span>(y)
<br />where P<span style="font-size:100%;"><sub>r</sub></span>(y) denotes the parity function which is 1 when y is odd and which is 0 when y is even.
<br /><b>
<br />
<br /></b></nobr> </p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Section4.1.htm#4.1%20%20%20%20Recursive%20Functions">Back to top</a></span> </p><p style="line-height: 150%;" align="left"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio3.gif" width="767" border="0" height="10" />
<br /><b> <nobr>
<br /><a name="4.1.3 Primitive Recursive Set">4.1.3 Primitive Recursive Set</a>
<br /></nobr></b>
<br />Any set R <span style="font-family:Symbol;">Í</span> N<sup>n</sup> defines a number - theoretic n-ary relation.
<br /><b><i>
<br />The characteristic function of a relation</i></b><i> R can be now defined as
<br />
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image160.gif" width="309" height="50" />
<br />
<br />Here R <span style="font-family:Symbol;">Í</span> N<sup>n</sup> and <x<sub>1</sub>, x<sub>2</sub>, . . ., x<sub>n</sub>><span style="font-family:Symbol;"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image18.gif" width="13" height="13" /> </span>N<sup>n</sup>.
<br />
<br />A relation R is said to be <b>primitive recursive</b> if its characteristic function is primitive recursive. The corresponding
<br /> predicate is also called primitive recursive.
<br /></i><b>
<br />
<br />
<br />Example 1:
<br /></b>Show that {<x,> / x <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image18.gif" width="13" height="13" /> N} which defines the relation of equality is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />Obviously f<x,> = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image155.gif" width="18" align="absmiddle" height="24" /> (| x – y |) defines a primitive recursive function such that f<x,> = 1 for x = y and otherwise
<br /> f<x> = 0.
<br />
<br />Thus f<x,> is the required characteristic function which is primitive recursive.
<br /><b>
<br />
<br />
<br />Example 2:
<br /></b>Show that for any fixed k the relation given by {<k,> / y > k} is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>sg(y <u> • </u> k) is the characteristic function of the required relation.
<br /><b>
<br />
<br />Example 3:
<br /></b>Show that the function f<x<sub>1</sub>, x<sub>2</sub>, y> defined as
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image161.gif" width="245" height="50" />
<br />
<br />Is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />The required function can be expressed as
<br />
<br />x<sub>2</sub> + (x<sub>1</sub> * y) * <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image155.gif" width="18" align="absmiddle" height="24" /> (x<sub>1 </sub><u> • </u> y).
<br /><i>
<br />
<br /></i> </p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Section4.1.htm#4.1%20%20%20%20Recursive%20Functions">Back to top</a></span> </p><p style="line-height: 150%;" align="left"> <i> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio4.gif" width="767" border="0" height="10" />
<br /></i> <nobr>
<br /><b><a name="4.1.4 Recursive Function">4.1.4 Recursive Function</a></b>
<br />
<br /><b>Regular Function:</b>
<br /><i>
<br />Let g <>1</sub>, x<sub>2</sub>,…, x<sub><span style="font-size:130%;">n</span></sub>, y > be a total function. If there exists atleast one value of y, say <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image162.gif" width="13" align="absmiddle" height="25" /><span style="font-family:Symbol;">Î</span> N, such that the function
<br /> g <>1</sub>, x<sub>2</sub>,…, x<sub>n</sub>,<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image163.gif" width="13" align="absmiddle" height="25" /> > = 0 for all n - tuples <>1</sub>, x<sub>2</sub>,…,x<sub>n</sub> > <span style="font-family:Symbol;">Î </span>N<sup>n</sup> then g is called a <b>regular function.
<br /></b></i>
<br />Not all total functions are regular, as can be seen from g <> = | y<sup>2 </sup>– x |. g <> is total, but |y<sup>2</sup>–x| = 0 for only those
<br /> values of x which are perfect squares and not for all values of x. This fact shows that there is no value of y <span style="font-family:Symbol;">Î </span>N such that
<br /> | y<sup>2 </sup>– x | = 0 for all x. On the other hand, the function y <u> • </u> x is regular because for y=0, y <u> • </u> x is zero for all x.
<br />
<br />
<br /><b>Minimization:</b>
<br />
<br /><i>A <b>function f <x<sub>1</sub>, x<sub>2</sub>,…, x<sub>n</sub>> is said to be defined from a total function</b> <b>g <>1</sub>, x<sub>2</sub>, ..., x<sub>n</sub>, y> by minimization or <span style="font-family:Symbol;">
<br /> m</span> operation</b> if
<br />
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image164.gif" width="456" height="48" />
<br />
<br />where <span style="font-family:Symbol;">m</span> <sub><span style="font-size:130%;">y</span></sub> means the least y greater than or equal to zero.
<br /></i>
<br />From the definition it follows that f <>1</sub>, x<sub>2</sub>, … , x<sub>n</sub> > is well-defined and total if g is regular. If g is not regular, then the
<br /> operation of minimization may produce a partial function.
<br />
<br /><b>
<br />Recursive Function:</b>
<br />
<br /><i>A function is said to be<b> recursive </b>iff it can be obtained from the initial functions by a finite number of applications of
<br /> the operations of composition, recursion, and minimization over regular functions</i>.
<br />
<br />It is clear from the definition that the set of recursive function properly includes the set of primitive recursive functions. Also the
<br /> set of recursive functions is closed under the operations of composition, recursion, and minimization over regular functions.
<br />
<br />
<br /><b>Partial Recursive Function:</b>
<br /><i>
<br />A function is said to be <b>partial recursive</b> iff it can be obtained from the initial functions by a finite number of
<br /> applications of the operations of composition, recursion, and minimization.
<br />
<br /></i><b>
<br />Example 1:</b>
<br />
<br />Show that the function f (x) = x/2 is a partial recursive function
<br />
<br /><b>Solution:</b>
<br />
<br />Let g <x,> = |2y – x|.
<br />
<br />The function g is not regular because |2y – x| = 0 only for even values of x.
<br />
<br />Define f(x) = <span style="font-family:Symbol;">m</span> <span style="font-size:100%;"><sub>y</sub></span> ( | 2y – x| ), then f (x) = x/2 for x even.
<br />
<br /><b>
<br />Example 2:
<br /></b>
<br />Let [<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image165.gif" width="26" align="absmiddle" height="22" />] be the greatest integer <sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image19.gif" width="13" align="absmiddle" height="16" /> </sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image165.gif" width="26" align="absmiddle" height="22" />. Show that [<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image165.gif" width="26" align="absmiddle" height="22" />] is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />Observe that (y + 1)<sup>2</sup> <u> • </u> x is zero for (y + 1)<sup>2 </sup> <span style="font-family:Symbol;">£</span> x and non-zero for (y + 1)<sup>2</sup> > x.
<br />
<br />Therefore, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image155.gif" width="18" align="absmiddle" height="24" /> ((y + 1)<sup>2 </sup><u> • </u> x) is 1 if (y + 1)<sup>2</sup> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image19.gif" width="13" height="16" /> x and cannot be equal to zero.
<br />
<br />The smallest value of y for which (y + 1)<sup>2</sup> > x is the required number [<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image166.gif" width="26" align="absmiddle" height="22" />] , hence
<br />
<br />[<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image166.gif" width="26" align="absmiddle" height="22" />] = <span style="font-family:Symbol;">m</span> <span size="3"><sub>y</sub></span>(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image155.gif" width="18" align="absmiddle" height="24" />((y + 1)<sup>2</sup> <u> • </u> x)) = 0.
<br />
<br />Note that [<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image166.gif" width="26" align="absmiddle" height="22" />] is defined for all x and hence is a recursive function.
<br />
<br />
<br /></nobr> </p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Section4.1.htm#4.1%20%20%20%20Recursive%20Functions">Back to top</a></span> </p><p style="line-height: 150%;" align="left"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio5.gif" width="767" border="0" height="10" />
<br /><b> <nobr>
<br /><a name="4.1.5 Recursive Sets">4.1.5 Recursive Sets</a>
<br /></nobr></b>
<br />A set A is called recursive (partial recursive) if its characteristic function <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image167.gif" width="25" align="absmiddle" height="22" /> is recursive (partial recursive). For any two sets
<br /> A and B with characteristic function <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image167.gif" width="25" align="absmiddle" height="22" /> and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image168.gif" width="24" align="absmiddle" height="22" /> , the characteristic functions of A<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image3.gif" width="17" height="13" /></sub>B and A<sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image48.gif" width="17" height="13" /></sub>B can also be obtained.
<br /><b>
<br />
<br />Example 1:
<br /></b>
<br />Show that the sets of even and odd natural numbers are both recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />The parity function is the required characteristic function for the set E of even natural numbers. Hence E is primitive recursive.
<br /> Also the set of odd natural numbers is ~E (complement of E), hence ~E is also primitive recursive.
<br /><b>
<br />
<br />Example 2:</b>
<br />
<br />Show that set of divisors of a positive integer n is recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />A number x <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image19.gif" width="13" height="16" /> n is a divisor of n iff | x * i - n| is equal to zero for one fixed value of i, 1 <span style="font-family:Symbol;">£</span><span style="font-family:Symbol;"> </span>i <span style="font-family:Symbol;">£ </span>n.
<br />
<br />This means that | x * i - n | is non zero for all 1 <span style="font-family:Symbol;">£</span> i <span style="font-family:Symbol;">£</span> n, if x is not a divisor.
<br />
<br />Therefore, the characteristic function of the required set is
<br />
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image169.gif" width="154" height="46" />
<br />
<br />where B denotes the set of divisors of n.
<br /><b>
<br />
<br />Note:
<br /></b>
<br />A predicate whose extension is a set of integers is said to be a number-theoretic predicate. Such a predicate is primitive
<br /> recursive (recursive) iff its extension is primitive recursive (recursive). The characteristic function of a predicate is the
<br /> characteristic function of its extension. If A is the extension of predicate P and <span style="font-family:Symbol;">Y</span><sub>P</sub> denotes the characteristic function of the
<br /> predicate P, then <span style="font-family:Symbol;">Y</span><sub>P</sub> = <span style="font-family:Symbol;">Y</span><sub>A
<br /></sub>
<br />For example, the predicates "is even" and "is a divisor of n" are recursive because their extensions are recursive sets.
<br />
<br />If A and B are the extensions of predicates P and Q respectively, then by definition.
<br /><span style="font-family:Symbol;">
<br />Y</span><sub>P </sub><span style="font-family:Symbol;"><sub>Ú</sub></span> <sub>Q</sub> = <span style="font-family:Symbol;">Y</span><sub>A<span style="font-family:Symbol;">È </span>B</sub> <span style="font-family:Symbol;">Y</span><sub>P </sub><span style="font-family:Symbol;"><sub>Ù</sub></span> <sub> Q</sub> = <span style="font-family:Symbol;">Y</span><sub>A <span style="font-family:Symbol;">Ç </span>B</sub> <span style="font-family:Symbol;">Y<sub>ù </sub></span><sub>P</sub> = <span style="font-family:Symbol;">Y</span><span style="font-size:180%;"><sub>~</sub></span> <sub> A
<br /></sub>
<br />It directly follows that if P and Q are recursive, then so are predicates P <span style="font-family:Symbol;">Ú </span> Q, P <span style="font-family:Symbol;">Ù </span> Q, and <span style="font-family:Symbol;">ù</span> P.
<br /><b>
<br />
<br />Example 3:
<br /></b>
<br />Let D (x) denote " number of divisors of x". Show that D(x) is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />We have shown that the function which defines the remainder upon division of y by x is primitive recursive. We shall denote
<br /> such a function by rm <x,y> . If a number x divides y, then the remainder is 0 and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image155.gif" width="18" align="absmiddle" height="24" /> (rm<x,>) = 1. Therefore the number of
<br /> divisors of y is given by
<br />
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Image170.gif" width="182" height="57" />
<br />
<br />This shows that D (y) is primitive recursive.
<br /><b>
<br />
<br />Example 4:
<br /></b>
<br />Show that the predicate "x is prime" is primitive recursive.
<br /><b>
<br />Solution:
<br /></b>
<br />A number x is a prime iff it has only two divisors 1 and x, also if it is not 1 or 0. Therefore, the characteristic function of the
<br /> extension of "x is not a prime" is
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio16.gif" width="320" border="0" height="43" />
<br />
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit4/Section4.1/Image/Sectio17.gif" width="452" border="0" height="35" />
<br /><b>
<br />
<br />Note:
<br /></b>
<br />In our discussion we considered only one induction variable in the definition of recursion. It is possible to consider two or more
<br /> induction variables. Note that in the definition of f<x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>,y> using recursion, x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub> were treated as parameters and
<br /> only y was treated as the induction variable. Now we define a function in which we have two induction variables and no
<br /> parameters.
<br />
<br />
<br />Consider Ackermann's function. The function A<x,> is defined by
<br />
<br />A<> = y + 1.
<br />A<> = A <>.
<br />A <> = A <>>.
<br />
<br />A <x,> is well defined and total. A <x,> is not primitive, but recursive.
<br /><b>
<br />
<br />Example 5:
<br /></b>
<br />Evaluate A <2,2> using the above definitions.
<br /><b>
<br />Solution:
<br /></b>
<br />A<> = A<>>
<br />A<> = A<>>
<br />A<2,> = A<>
<br />A<> = A<> >
<br /> = A<>>
<br />
<br />A<> = 2
<br />A<> = A<>
<br /> = 3
<br />
<br />A<> = A<>
<br /> = A<>>
<br />A<> = A<>>
<br /> = A<>
<br /> = 4
<br />
<br />A<> = A<>
<br /> = 5
<br />
<br />A<> = A<>
<br /> = A<>>
<br />
<br />A<> = A<>>
<br />A<> = A<>>
<br /> = A<>
<br /> = 5
<br />
<br />A<> = A<>
<br /> = 6
<br />
<br />A <> = A <>
<br /> = 7.
<br /><b>
<br />Algorithm Prime:
<br /></b>
<br />Given an integer i greater than 1, this algorithm will determine whether the integer is a prime number. Note that the only divisors
<br /> of a prime number are 1 and the number itself.
<br /></p><ol><li> Set j <span style="font-family:Symbol;">¬ </span>2.</li><li> If j <span style="font-family:Symbol;">³</span> i then output " i is prime" and Exit.</li><li> If j <span style="font-family:Symbol;">/</span> i then output " i is not prime" and Exit.</li><li> Set j <span style="font-family:Symbol;">¬ </span>j + 1, goto step 2.</li></ol>
<br /><b>
<br />Algorithm Perfect:
<br /></b>
<br />This algorithm decides whether there exists a perfect number greater than some integer i. Consider the set of all divisors of a
<br /> number except the number itself. A perfect number is one whose sum of all such divisors equals the number. The number 6 is a
<br /> perfect number since 1 + 2 + 3 = 6.
<br /><ol><li> Set k <span style="font-family:Symbol;">¬</span> i.</li><li> Set k <span style="font-family:Symbol;">¬ </span>k + 1.</li><li> Set SUM <span style="font-family:Symbol;">¬ </span>0.</li><li> Set j <span style="font-family:Symbol;">¬</span>1.</li><li> If j <><li> If SUM = k then output k and Exit, otherwise go to step 2.</li><li> If j / k then set SUM <span style="font-family:Symbol;">¬ </span>SUM + j. Set j <span style="font-family:Symbol;">¬ </span>j + 1 and go to step 5</li></ol>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com2tag:blogger.com,1999:blog-7399621945608861143.post-89929854051770969292008-08-12T04:26:00.000-07:002008-12-23T01:55:02.554-08:00Rings and fields<b><nobr><a name="3.5.1 Rings"></a>
<br /> </nobr> </b> <div style="left: 12px; top: 16px; width: 669px; height: 19px; position: absolute;"><nobr>
<br /><i>An algebraic system with two binary operations + and •, <s,> is called a <b>ring</b> if</i>
<br /><ol><li> <s,> is an abelian group.</li><li> <s,> is a semigroup.</li><li> The operation • is distributive over + that is, for any a, b, c <span style="font-family:Symbol;">Î</span> S, a • (b + c) = a • b + a • c and
<br /> (b + c) • a = b • a + c • a.</li></ol> <b>
<br />Note:</b>
<br />
<br />If <s,> is commutative, then <s,> is called a commutative ring.
<br />
<br />Similarly if <s,> is a monoid, then <s,> is called a ring with identity.
<br />
<br />
<br /> Familiar examples of rings are the set of integers, real numbers, rational numbers, and complex numbers under the operations
<br /> of addition and multiplication.
<br />
<br />
<br />The additive identity is denoted by 0 and multiplication identity by 1.
<br />
<br />
<br /> <i> If <s,> is a ring and for any a, b <span style="font-family:Symbol;">Î</span> S such that a <span style="font-family:Symbol;">¹</span> 0, b <span style="font-family:Symbol;">¹</span> 0, a• b <span style="font-family:Symbol;">¹</span> 0 then <s,> is a ring without zero divisors. A communicative ring with identity and without divisors of zero is called an<b> integral domain</b>.
<br /> </i>
<br />
<br /></nobr> <p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.5/Section3.5.htm#3.5%20Rings%20and%20Fields">Back to top</a></span></p> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.5/Image/Sectio2.gif" width="767" border="0" height="10" />
<br /> <nobr>
<br />
<br /> <a name="3.5.2 Field">3.5.2 Field</a>
<br /> </nobr></b>
<br /><i>A commutative ring <s,> which has more than one element such that every non-zero element of S has a
<br /> multiplicative inverse in S is called a <b>field</b>.
<br /> </i> <b>
<br />Note:
<br /> </b>
<br />The ring of integers is an integral domain, which is not a field.
<br />
<br />The rings of real numbers and rational numbers are examples of fields.
<br /> <b>
<br />
<br /> Example 1:
<br /> </b>
<br />The algebraic system <z<sub>n</sub>, +<sub>n</sub>, *<sub>n</sub>> consisting of equivalence classes generated by the relation congruence modulo n for some
<br /> fixed integer n over the set of integers is a ring.
<br />
<br />
<br />
<br />Note that <z<sub>6</sub>, +<sub>6</sub>, *<sub>6</sub>> is not an integral domain since [3] * [2] = [0]. On the other hand <z>7</sub>, + <sub>7</sub>,<sub> </sub>* <sub>7</sub>> is an integral domain.
<br /> In fact, <z<sub>n</sub>, +<sub>n</sub>, *<sub>n</sub>> is a field if and only if n is a prime integer.
<br />
<br /> <b>
<br />Example 2:
<br /> </b>
<br />Let S be a set and <i>P</i>(S) its power set. On <i>P</i>(S) we define operations + and • as follows
<br />
<br />A + B = {x<span style="font-family:Symbol;">Î</span> S / x<span style="font-family:Symbol;">Î</span> A <span style="font-family:Symbol;">È</span> B and x<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.5/Image/Image79.gif" width="13" align="absmiddle" height="16" />A<span style="font-family:Symbol;">Ç</span> B}
<br />
<br />A • B = A<span style="font-family:Symbol;">È</span> B for all A, B <span style="font-family:Symbol;">Î</span> <i>P</i>(S)
<br />
<br />Then it is easy to verity that <<i>P</i>(S), +, •> is a ring called the ring of subsets of S.
<br /> <i>
<br />A subset R <span style="font-family:Symbol;">Í</span> S ,where <s,> is a ring, is called a <b>subring</b> if <r,+> is itself a ring with operations + and •
<br /> restricted to R.
<br /> </i> <b>
<br />Example:
<br /> </b>
<br />The ring of even integers is a subring of the ring of integers.
<br />
<br />
<br /> <i>
<br />Let <r,>•<i>> and <s,<span style="font-family:Symbol;">Å</span> , <span style="font-family:Wingdings;">¤</span> > be rings. A mapping of g : R<span style="font-family:Symbol;">®</span> S is called a <b>ring homomorphism</b> from <r,> and
<br /> <s, face="Symbol">Å</span> ,<span style="font-family:Wingdings;">¤</span> > if for any a, b, <span style="font-family:Symbol;">Î</span> R
<br /> </i>
<br />g(a + b) = g(a) <span style="font-family:Symbol;">Å</span> g(b) and g(a • b) = g(a) <span style="font-family:Wingdings;">¤</span> g(b).
<br /> <b>
<br />
<br /> Note:
<br /> </b>
<br />The first condition is a group homomorphism from <r,> to <s, face="Symbol">Å</span> > and second is a semigroup homomorphism form <r,>
<br /> to <s, face="Wingdings">¤</span> >.
<br />
<br />
<br /> <b> Property:
<br /> </b>
<br />The distributive property is preserved by ring homomorphism.
<br /> <b>
<br />Proof:
<br /> </b>
<br />Let <r,> and <s, face="Symbol">Å</span> , <span style="font-family:Wingdings;">¤</span> > be rings and g : R<span style="font-family:Symbol;">®</span> S be a ring homomorphism.
<br />
<br />For any a, b, c <span style="font-family:Symbol;">Î</span> R,
<br />
<br />g[a • (b + c)] = g(a) <span style="font-family:Wingdings;">¤</span> g(b + c)
<br /> = g(a) <span style="font-family:Wingdings;">¤</span> [g(b) <span style="font-family:Symbol;">Å</span> g(c)]
<br /> = [g(a) <span style="font-family:Wingdings;">¤</span> g(b)] <span style="font-family:Symbol;">Å</span> [g(a) <span style="font-family:Wingdings;">¤</span> g(c)]
<br /> = g(a • b + a • c)
<br />
<br />which proves the property</div>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-19944278334609223242008-08-12T04:25:00.004-07:002008-12-23T01:55:02.554-08:00Normal Subgroups <b><a name="Normal Subgroups"></a></b><i>
<br />A subgroups <h,> of <g,> is called a <b>normal subgroup</b> if for any a<span style="font-family:Symbol;">Î</span> G, aH = Ha.
<br /> </i><b>
<br />Note 1:</b>
<br />
<br />aH = Ha does not necessarily mean that a * h = h * a for every h <span style="font-family:Symbol;">Î</span> H. It only means that a * h<sub>i</sub> = hj * a for some h<sub>i</sub>, hj <span style="font-family:Symbol;">Î</span> H.
<br /> <b>
<br />Note2:
<br /> </b>
<br />Every subgroup of an abelian group is normal.
<br />
<br /><b>Note3:
<br /> </b>
<br />The trivial subgroups are also normal.
<br /> <b>
<br />Note4:
<br /> </b>
<br />If H is a normal subgroup, then both the left and right costs of H in G are equal.
<br /> <b>
<br />
<br />
<br />Example 1:
<br /> </b>
<br />Consider the symmetric group <s<sub>3</sub>, <span style="font-family:Symbol;">à</span>>.
<br /> <b>
<br />Solution:
<br /> </b>
<br />From the Table3 it is clear that {p<sub>1</sub>, p<sub>2</sub>}, {p<sub>1</sub>, p<sub>3</sub>}, {p<sub>1</sub>, p<sub>4</sub>} and {p<sub>1</sub>, p<sub>5, </sub>p<sub>6</sub>} are subgroups of <s<sub>3</sub>, <span style="font-family:Symbol;">à</span>>. The left cosets of
<br /> {p<sub>1</sub>, p<sub>2</sub>} are {p<sub>1</sub>, p<sub>2</sub>}, {p<sub>3</sub>, p<sub>6</sub>} and {p<sub>4</sub>, p<sub>5</sub>}, while the right cosets of {p<sub>1</sub>, p<sub>2</sub>} are {p<sub>1</sub>, p<sub>2</sub>}, {p<sub>3</sub>, p<sub>5</sub>} and {p<sub>4</sub>, p<sub>5</sub>} and
<br /> hence {p<sub>1</sub>, p<sub>2</sub>} is not a normal subgroup. Similarly, {p<sub>1</sub>, p<sub>3</sub>}, {p<sub>1</sub>, p<sub>4</sub>} are not normal subgroups.
<br />
<br />But the left and right cosets of {p<sub>1</sub>, p<sub>5</sub>, p<sub>6</sub>} are {p<sub>1</sub>, p<sub>5</sub>, p<sub>6</sub>} and {p<sub>2</sub>, p<sub>3</sub>, p<sub>4</sub>}.
<br />
<br />Hence, {p<sub>1</sub>, p<sub>5</sub>, p<sub>6</sub>} is a normal subgroup.
<br />
<br />
<br /> <b>
<br />Theorem 3.4-1:
<br /> </b>
<br />A subgroup H is normal in G if and only if for all a <span style="font-family:Symbol;">Î</span> H and x <span style="font-family:Symbol;">Î</span> G, xax<sup><span style="font-size:130%;">-</span>1</sup> <span style="font-family:Symbol;">Î</span> H.
<br /> <b>
<br />Proof:
<br /> </b>
<br />Given that H is normal, therefore xH = Hx, for all x <span style="font-family:Symbol;">Î</span> G.
<br />
<br />Therefore, xHx <span style="font-size:100%;"><sup>-</sup></span><sup>1</sup> = Hx * x <sup><span style="font-size:100%;">-</span></sup><sup>1</sup>.
<br />
<br />That is, xHx <sup><span style="font-size:100%;">-</span></sup><sup>1</sup> = H.
<br />
<br />Hence, for all a<span style="font-family:Symbol;">Î</span> H and x <span style="font-family:Symbol;">Î</span> G, xax <sup><span style="font-size:100%;">-</span></sup><sup>1</sup><span style="font-family:Symbol;">Î</span> <sup> </sup>H.
<br />
<br />Conversely, suppose for all a <span style="font-family:Symbol;">Î</span> H and x <span style="font-family:Symbol;">Î</span> G, xax <span style="font-size:100%;"><sup>-</sup></span><sup>1</sup><span style="font-family:Symbol;">Î</span> <sup> </sup>H, then xHx <sup><span style="font-size:100%;">-</span></sup><sup>1 </sup> <span style="font-family:Symbol;">Í</span> <sup> </sup>H …… (1)
<br />
<br />Thus, xH <span style="font-family:Symbol;">Í</span> Hx, for all x <span style="font-family:Symbol;">Î</span> G ……(2)
<br />
<br />By replacing x by x <sup><span style="font-size:100%;">-</span></sup><sup>1</sup> in (1), we have x <sup><span style="font-size:100%;">-</span></sup><sup>1</sup>Hx <span style="font-family:Symbol;">Í</span> H.
<br />
<br />Thus, Hx <span style="font-family:Symbol;">Í</span> xH, for all x <span style="font-family:Symbol;">Î</span> G …… (3)
<br />
<br />From (2) and (3) we have xH = Hx.
<br />
<br />Hence, H is normal.
<br /> <b>
<br />
<br />
<br />Theorem 3.4-2:
<br /> </b>
<br />Let <g,> and <h, face="Symbol">D</span>> be groups and g : G <span style="font-family:Symbol;">®</span> H be a homomorphism. Then the kernel of g is a normal subgroup.
<br /> <b>
<br />Proof:</b>
<br />
<br />We know that ker(g) is a subgroup of <g,> Now for any a <span style="font-family:Symbol;">Î</span> G, and k <span style="font-family:Symbol;">Î</span> ker(g),
<br />
<br />g(a<sup>-1 </sup>* k * a) = g(a <sup>-1</sup>) <span style="font-family:Symbol;">D</span> g (k) <span style="font-family:Symbol;">D</span> g(a).
<br /> = g(a <sup>-1</sup>) <span style="font-family:Symbol;">D</span> e<sub>H </sub> <span style="font-family:Symbol;">D</span> g(a).
<br /> = [g(a)]<sup>-1 </sup><span style="font-family:Symbol;">D</span> g(a) = e<sub>H </sub> .
<br />
<br />Hence, a<sup>-1</sup> * k * a <span style="font-family:Symbol;">Î</span> ker(g).
<br />
<br />That is, ker(g) is a normal subgroup of <g,>.
<br />
<br />
<br /> <b>
<br />Theorem 3.4-3:
<br /> </b>
<br />Let H be a normal subgroup of a group (G, *), then the relation "a <span style="font-family:Symbol;">º</span> b(mod H)" is a congruence relation, where a <span style="font-family:Symbol;">º</span> b(mod H)
<br /> iff b<sup>-1 </sup>* a <span style="font-family:Symbol;">Î</span> H.
<br /> <b>
<br />Proof:
<br /> </b>
<br />From Step 1 of the Lagrange’s Theorem, it is clear that the relation a <span style="font-family:Symbol;">º</span> b(mod H) is an equivalence relation .
<br />
<br />Now we shall prove that a <span style="font-family:Symbol;">º</span> b(mod H) is congruence.
<br />
<br />That is, we’ll prove that
<br />
<br />if a <span style="font-family:Symbol;">º</span> a<span style="font-family:Symbol;">¢</span> (mod H) and b <span style="font-family:Symbol;">º</span> b<span style="font-family:Symbol;">¢</span> (mod H) then a * b <span style="font-family:Symbol;">º</span> a<span style="font-family:Symbol;">¢</span> * b<span style="font-family:Symbol;">¢</span> (mod H).
<br />
<br />Let a <span style="font-family:Symbol;">º</span> a<span style="font-family:Symbol;">¢</span> (mod H) and b <span style="font-family:Symbol;">º</span> b<span style="font-family:Symbol;">¢</span> (mod H).
<br />
<br />Then by the definition of the relation,
<br />
<br />(a<span style="font-family:Symbol;">¢</span> )<sup>-1 </sup>* a <span style="font-family:Symbol;">Î</span> H and (b<span style="font-family:Symbol;">¢</span> )<sup>-1 </sup>* b <span style="font-family:Symbol;">Î</span> H. Therefore,
<br />
<br />(a<span style="font-family:Symbol;">¢</span> * b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * (a * b) = ((b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * (a<span style="font-family:Symbol;">¢</span> )<sup>-1</sup>) * (a * b)
<br /> = (b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * ((a<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * a) * (b<span style="font-family:Symbol;">¢</span> * (b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup>) * b
<br /> = ((b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * ((a<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * a) * b<span style="font-family:Symbol;">¢</span> ) * (b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * b
<br />
<br />Since H is normal and (a<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * a <span style="font-family:Symbol;">Î</span> H, we have (b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup>* ((a<span style="font-family:Symbol;">¢</span> )<sup>-1</sup>*a) * b <span style="font-family:Symbol;">Î</span> H.
<br />
<br />Also, (b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * b <span style="font-family:Symbol;">Î</span> H, we have ((b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * ((a<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * a) * b<span style="font-family:Symbol;">¢</span> ) * ((b<span style="font-family:Symbol;">¢</span> )<sup>-1</sup> * b)<span style="font-family:Symbol;">Î</span> H.
<br />
<br />By definition of a <span style="font-family:Symbol;">º</span> b(mod H), we have a * b <span style="font-family:Symbol;">º</span> (a<span style="font-family:Symbol;">¢</span> * b<span style="font-family:Symbol;">¢</span> )(mod H).
<br />
<br />Hence, <span style="font-family:Symbol;">º</span> (mod H) is a congruence relation.
<br />
<br />
<br />
<br />Let G be a group and let H be a subgroup of G, then the relation a <span style="font-family:Symbol;">º</span> b (mod H) is an equivalence relation. We denote
<br /> G/H ={[a] / a <span style="font-family:Symbol;">Î</span> G}(the set of all equivalence classes defined by the relation "<span style="font-family:Symbol;">º</span> (mod H)"). As each equivalence class [a] is
<br /> nothing but a left coset aH, we have G/H = {aH / a <span style="font-family:Symbol;">Î</span> G}. G/H is called <b><i>the quotient set of G defined by H</i></b>. If the subgroup
<br /> H is normal, then by Theorem 3.4-3 the relation "<span style="font-family:Symbol;">º</span> (mod H)" is a congruence relation.
<br />
<br />Therefore we can define a binary relation <span style="font-family:Symbol;">Ä</span> on G/H, by aH <span style="font-family:Symbol;">Ä</span> bH = (a * b)H.
<br />
<br />The operation <span style="font-family:Symbol;">Ä</span> is well defined since <span style="font-family:Symbol;">º</span> (mod H) is a congruence relation.
<br />
<br />[Suppose aH is equivalent to cH then, aH <span style="font-family:Symbol;">Ä</span> bH = cH <span style="font-family:Symbol;">Ä</span> bH, a coset can be replaced by an equivalent coset in the process of
<br /> operation. By doing so the result will not be changed and remains the same.]
<br /> <b>
<br />
<br />
<br />Theorem 3.4-4:
<br /> </b>
<br />Let H be a normal subgroup of a group (G, *), then (G/H, <span style="font-family:Symbol;">Ä</span> ) is a group, called <b><i>factor group</i></b>, where aH <span style="font-family:Symbol;">Ä</span> bH = (a * b)H.
<br /> Also, there exists a natural homomorphism from (G, *) onto (G/H, <span style="font-family:Symbol;">Ä</span> ).
<br /> <b>
<br />Proof:
<br /> </b>
<br />We have already seen that the operation <span style="font-family:Symbol;">Ä</span> is well defined.
<br />
<br />Now, (aH <span style="font-family:Symbol;">Ä</span> bH) <span style="font-family:Symbol;">Ä</span> cH = ((a * b)H) *cH
<br /> = ((a * b) * c)H
<br /> = (a * (b * c))H
<br /> = aH <span style="font-family:Symbol;">Ä</span> (b * c)H
<br /> = aH <span style="font-family:Symbol;">Ä</span> (bH <span style="font-family:Symbol;">Ä</span> cH)
<br />
<br />Thus, <span style="font-family:Symbol;">Ä</span> is associative.
<br />
<br />Further, eH = H acts as an identity element of G/H.
<br />
<br />For, H <span style="font-family:Symbol;">Ä</span> aH = (e * a)H = aH = (a * e)H = aH <span style="font-family:Symbol;">Ä</span> eH.
<br />
<br />Also, for aH <span style="font-family:Symbol;">Î</span> G/H, there exists a<sup>-1</sup>H such that aH <span style="font-family:Symbol;">Ä</span> a <sup>-1</sup>H = (a * a <sup>-1</sup>)H = eH.
<br />
<br />Thus, for every aH <span style="font-family:Symbol;">Î</span> G/H, there exists inverse a <sup>-1</sup>H in G/H.
<br />
<br />Hence (G/H, <span style="font-family:Symbol;">Ä</span> ) is a group.
<br />
<br />Define g: G <span style="font-family:Symbol;">®</span> G/H , by g(a) = aH.
<br />
<br />Then, g(a * b) = (a * b)H
<br /> = aH <span style="font-family:Symbol;">Ä</span> bH
<br /> = g(a) <span style="font-family:Symbol;">Ä</span> g(b)
<br />
<br />Thus, g is a homomorphism.
<br />
<br />For each aH <span style="font-family:Symbol;">Î</span> G/H, there exists a <span style="font-family:Symbol;">Î</span> G such that g(a) = aH.
<br />
<br />Thus, g is onto.
<br />
<br />Hence, the theorem.
<br />
<br />
<br /> <b>
<br />Theorem 3.4-5 (First Isomorphism Theorem):
<br /> </b>
<br />Let g be a homomorphism from a group (G , *) to a group (H , <span style="font-family:Symbol;">D</span> ) and let K be the kernel of g and H<span style="font-family:Symbol;">¢</span> <span style="font-family:Symbol;">Í</span> H be the image set of G under the mapping g. Then G/H is isomorphic to H<span style="font-family:Symbol;">¢</span> .
<br /> <b>
<br />Proof:
<br /> </b>
<br />Since K is a kernel of a homomorphism it must be a normal subgroup of G.
<br />
<br />By Theorem 3.4-4, the mapping f from (G , *) into the factor group (G/H , <span style="font-family:Symbol;">Ä</span> ) defined by f(a) = aK,
<br />for all a <span style="font-family:Symbol;">Î</span> G is an epimorphism.
<br />
<br />Now we define h : G/H <span style="font-family:Symbol;">®</span> H<span style="font-family:Symbol;">¢</span> such that h(aK) = g(a).
<br />
<br />Now we have the following structure of mapping.
<br /> <span style="font-size:85%;">
<br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.4/Image/nor.gif" width="280" border="0" height="151" />
<br /> </span> <b>
<br />Claim 1:</b> <i>h is well defined.
<br /> </i>
<br />Suppose aK =bK for some a, b <span style="font-family:Symbol;">Î</span> G, then a*k<sub>1</sub> = b*k<sub>2, </sub>for some k<sub>1</sub>, k<sub>2</sub> <span style="font-family:Symbol;">Î</span> K.
<br />
<br />Then, g(a * k<sub>1</sub>) = g(a) <span style="font-family:Symbol;">D</span> g(k<sub>1</sub>)
<br /> = g(a) <span style="font-family:Symbol;">D</span> e<span style="font-family:Symbol;">¢</span>
<br /> = g(a)
<br />
<br />Also, g(a * k<sub>1</sub>) = g(b * k<sub>2</sub>)
<br /> = g(b) <span style="font-family:Symbol;">D</span> g(k<sub>2</sub>)
<br /> = g(b) <span style="font-family:Symbol;">D</span> e<span style="font-family:Symbol;">¢</span>
<br /> = g(b).
<br />
<br />Thus, g(a) = g(b).
<br />
<br />That is, h(aK) = h(bK).
<br />
<br />Hence, the claim.
<br /> <b>
<br />
<br />
<br />Claim 2:</b> <i>h is a homomorphism.
<br /> </i>
<br />h(aK <span style="font-family:Symbol;">Ä</span> bK) = h((a * b)K)
<br /> = g(a * b)
<br /> = g(a) <span style="font-family:Symbol;">D</span> g(b)
<br /> = h(aK) <span style="font-family:Symbol;">D</span> h(bK), for all aK, bK <span style="font-family:Symbol;">Î</span> G/H.
<br />
<br />Hence, h is a homomorphism.
<br />
<br />
<br />
<br /><b>Claim 3:</b> <i>h is both one-one and onto.
<br /> </i>
<br />For y <span style="font-family:Symbol;">Î</span> H<span style="font-family:Symbol;">¢</span> , there exists a <span style="font-family:Symbol;">Î</span> G such that g(a) = y, since g is onto.
<br />
<br />Hence, for each y <span style="font-family:Symbol;">Î</span> H<span style="font-family:Symbol;">¢</span> , there exists a <span style="font-family:Symbol;">Î</span> G such that y = g(a) = h(aK).
<br />
<br />Thus, h is onto.
<br />
<br />Suppose h(aK) = h(bK).
<br />
<br />Then, g(a) = g(b).
<br />
<br />Therefore, (g(b))<sup>-1</sup><span style="font-family:Symbol;">D</span> <sup> </sup>g(a) = e<span style="font-family:Symbol;">¢</span> .
<br />
<br />i.e., g(b<sup>-1 </sup>* a) = e<span style="font-family:Symbol;">¢</span> .
<br />
<br />i.e., b<sup>-1</sup> * a <span style="font-family:Symbol;">Î</span> K.
<br />
<br />i.e., aK = bK (by definition of the relation a <span style="font-family:Symbol;">º</span> b(mod K) )
<br />
<br />Hence, h is one-one.
<br />
<br />Hence, h is an isomorphism from G/H onto H<span style="font-family:Symbol;">¢</span> .
<br /> <b>
<br />
<br /> </b><i> Let (G , *) and (H , <span style="font-family:Symbol;">D</span> ) be two groups. The direct product of these two groups is the algebraic structure (G </i><span style="font-family:Symbol;">´</span> <i> H , </i><b>·</b><i>) in
<br /> which the binary operation </i><b>·</b><i> on G</i><span style="font-family:Symbol;">´</span> <i> H is given by</i> <i> (g<sub>1</sub> , h<sub>1</sub>) </i><b>·</b><i> (g<sub>2</sub> , h<sub>2</sub>) = (g<sub>1</sub>*g<sub>2</sub> , h<sub>1</sub><span style="font-family:Symbol;">D</span> <sub> </sub>h<sub>2</sub>), for any
<br /> (g<sub>1</sub> , h<sub>1</sub>) , (g<sub>2</sub> , h<sub>2</sub>) <span style="font-family:Symbol;">Î</span> G</i><span style="font-family:Symbol;">´</span> <i> H.
<br />
<br />It can be shown that (G </i><span style="font-family:Symbol;">´</span> <i> H , </i><b>·</b><i>) is a group and (e<sub>G</sub> , e<sub>H</sub>) is the identity element of (G </i><span style="font-family:Symbol;">´</span> <i> H , </i><b>·</b><i>) and inverse of any element
<br /> (g , h) is (g <sup>-1</sup> , h <sup>-1</sup>).</i>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1tag:blogger.com,1999:blog-7399621945608861143.post-71391296019712749202008-08-12T04:25:00.003-07:002008-12-23T01:55:02.554-08:00Cosets and Lagrange's Theorem<b><a name="Cosets and Lagrange's Theorem"></a></b><br />In this section we prove a very important theorem, popularly called "Lagrange’s Theorem", which had influenced to initiate the<br /> study of an important area of group theory called "Finite Groups". Finite groups have great applications in the study of finite<br /> geometrical and combinational structures.<br /> <br />Let (G, *) be a group and let H be a subgroup G.<br /> <i> A <b>left coset</b> corresponding to an element a<span style="font-family:Symbol;">Î</span> G, denoted by aH is the set</i> <i>{a * H / h <span style="font-family:Symbol;">Î</span> H}.<br /> Similarly, a <b>right coset</b> corresponding to an element a<span style="font-family:Symbol;">Î</span> G, denoted by Ha is the set </i> <i>{h * a / h <span style="font-family:Symbol;">Î</span> H}.</i><br /> <br /><br /> <br /><b>Example:<br /> </b> <br />For the group, (Z<sub>4</sub>, +) = ({[0], [1], [2], [3]} , +) and for the subgroup H = {[0], [2]}, we have the following left cosets.<br /> <br />[3] + H = {[3], [1]}.<br /> [2] + H = {[2], [0]}= H.<br /> [1] + H = {[1], [3]}.<br /> [0] + H = {[0], [2]} = H.<br /> <br /><br /> <br />Now we shall prove our main result, the Lagrange’s theorem.<br /> <span style="font-size:100%;"> <b> <br />Theorem [Lagrange’s Theorem]:<br /> </b></span> <br />In a finite group order of any subgroup divides the order of the group.<br /> <b> <br />Proof:<br /> <br />Step 1: </b><i>Let (G, *) be a group and let H be a subgroup of G. </i> <i>Define, for all a, b<span style="font-family:Symbol;">Î</span> G, a <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.3/Images/Image75.gif" width="13" align="absbottom" height="12" /> b(mod H) if and only if<br /> b<sup>-1</sup> * a <span style="font-family:Symbol;">Î</span> H. Then the </i> <i>relation, <span style="font-family:Symbol;">º</span> (modH), "congruence modulo H", is an equivalence relation.<br /> </i> <br />For all a <span style="font-family:Symbol;">Î</span> G, we have, a<sup>-1</sup> * a = e <span style="font-family:Symbol;">Î</span> H.<br /> <br />So, a <span style="font-family:Symbol;">º</span> a (mod H)<br /> <br />Thus, <span style="font-family:Symbol;">º</span> is reflexive.<br /> <br />Let a <span style="font-family:Symbol;">º</span> b (mod H).<br /> <br />Then, b<sup>-1</sup> * a <span style="font-family:Symbol;">Î</span> H.<br /> <br />So, (b<sup>-1</sup> * a) <sup>–1</sup> <span style="font-family:Symbol;">Î</span> H.<br /> <br />That is, a<sup>-1</sup> * (b<sup>-1</sup>)<sup>-1</sup> <span style="font-family:Symbol;">Î</span> H.<br /> <br />That is, a<sup>-1</sup> * b <span style="font-family:Symbol;">Î</span> H.<br /> <br />So, by definition of "<span style="font-family:Symbol;">º</span> ", b <span style="font-family:Symbol;">º</span> a (mod H)<br /> <br />Thus, <span style="font-family:Symbol;">º</span> is symmetry.<br /> <br />Let a <span style="font-family:Symbol;">º</span> b (mod H) and b <span style="font-family:Symbol;">º</span> c(mod H).<br /> <br />Then, b<sup>-1</sup> * a <span style="font-family:Symbol;">Î</span> H and c<sup>-1</sup> * b <span style="font-family:Symbol;">Î</span> H<br /> <br />Therefore, (c<sup>-1</sup> * b) * (b<sup>-1</sup> * a) <span style="font-family:Symbol;">Î</span> H.<br /> <br />That is, c<sup>-1</sup> * a <span style="font-family:Symbol;">Î</span> H<br /> <br />So, a <span style="font-family:Symbol;">º</span> c (mod H).<br /> <br />Implies, " <span style="font-family:Symbol;">º</span> " is transitive.<br /> <br />Hence, the step 1.<br /> <br /><br /> <b> <br />Step 2: </b><i>For a<span style="font-family:Symbol;">Î</span> G, the equivalence class [a] is nothing but the left coset a * H. </i> <i>Further G is partitioned into distinct<br /> cosets.<br /> </i> <br />By definition, [a] = { x <span style="font-family:Symbol;">Î</span> G / x <span style="font-family:Symbol;">º</span> a (mod H) }<br /> = { x <span style="font-family:Symbol;">Î</span> G / a<sup>-1</sup> * x <span style="font-family:Symbol;">Î</span> H}<br /> = {x <span style="font-family:Symbol;">Î</span> G / h = a<sup>-1</sup> * x, for h <span style="font-family:Symbol;">Î</span> H}<br /> = { x <span style="font-family:Symbol;">Î</span> G / a * h = x, for h <span style="font-family:Symbol;">Î</span> H}<br /> = { a * h / for h <span style="font-family:Symbol;">Î</span> H}<br /> = a * H.<br /> <br />Thus, each equivalence class [a] is the left coset a * H of G corresponding to ‘a’.<br /> <br />Since " <span style="font-family:Symbol;">º</span> " is an equivalence relation G, distinct equivalence classes partitions the whole group.<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.3/Images/Cos.gif" width="155" border="0" height="97" /><br /> <br />Hence the Step 2.<br /> <br /><br /> <b> <br />Step 3: </b><i>Let G be a group and let H be a subgroup of H. Then, | a * H | = | H |.<br /> </i><br />Let h<sub>i</sub> and h<sub>j</sub> <span style="font-family:Symbol;">Î H </span>be such that h<sub>i</sub> <span style="font-family:Symbol;">¹</span> h<sub>j</sub> . Suppose a * h<sub>i</sub> = a * h<sub>j</sub>.<br /> <br />Then, a <sup>-1 </sup>* (a * h<sub>i</sub>) = a<sup>-1</sup> * ( a* h<sub>j</sub>).<br /> <br />That is, (a <sup>-1 </sup>* a) * h<sub>i</sub> = (a<sup>-1</sup> * a)* h<sub>j<br /> </sub> <br />That is, e * h<sub>i</sub> = e * h<sub>j </sub>.<br /> <br />That is, h<sub>i</sub> = h<sub>j</sub> .<br /> <br />A contradiction to the fact that h<sub>i</sub> <span style="font-family:Symbol;">¹</span> h<sub>j</sub>.<br /> <br />Thus, if h<sub>i</sub> and h<sub>j</sub> are distinct in H, then a * h<sub>i</sub> and a* h<sub>j</sub> are distinct in a * H.<br /> <br />Hence, | a * H | = | H |.<br /> <br /><br /> <b> <br />Step 4: </b><i>Let G be a finite group and let H be a subgroup then o(H)</i><span style="font-family:Symbol;">½</span><i>o(G).<br /> </i> <br />By Step2, G = a<sub>1</sub> * H <span style="font-family:Symbol;">È</span> a<sub>2</sub> * H <span style="font-family:Symbol;">È</span> … <span style="font-family:Symbol;">È</span> a<sub>t</sub> * H, where,<br /> <br />a<sub>1</sub> * H, a<sub>2</sub> * H ,…, a<sub>t</sub> * H are the distinct coset of G ( and have no common elements)<br /> <br />Therefore, |G| = |a<sub>1</sub>*H| + |a<sub>2</sub> * H| + … + |a<sub>t</sub> * H|<br /> <br />By Step 3, |G| = |H| + |H| + … + |H| (t times)<br /> <br />So, |G| = t |H|<br /> <br />That is, |H| <span style="font-family:Symbol;">½</span> |G| = t (is an integer).<br /> <br />Hence the order of H divides the order of G.<br /> <br />Hence the Lagrange’s Theorem.<br /> <br /><br /> <br /><i>The number of left cosets of H in a group G in called the <b>index</b> of H in G.</i><br /> <br />From the Lagrange’s Theorem for a subgroup H, we have the index k of G given by<br /> <br />k = | H | <span style="font-family:Symbol;">½</span> | G |<br /> <b> <br /><br /> <br />Corollary:<br /> </b> <br />If (G, *) is a finite group of order n, then for any a <span style="font-family:Symbol;">Î</span> G, a<sup>n</sup> = e, where e is the identity of the group G. (Prove !)Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-31424273693776702062008-08-12T04:25:00.001-07:002008-12-23T01:55:02.554-08:00Subgroups<b><a name="Subgroups"></a></b><br /> <br /><i>Let <g,> be a group and S <span style="font-family:Symbol;">Î </span> G be such that it satisfies the following conditions : </i><br /> <ol type="i"><li> e <span style="font-family:Symbol;">Î </span>S, where e is the identity of <g,></li><li> For any a <span style="font-family:Symbol;">Î </span>S, a<sup>-1</sup> <span style="font-family:Symbol;">Î </span>S.</li><li> For a, b <span style="font-family:Symbol;">Î </span>S, a * b <span style="font-family:Symbol;">Î</span> S.</li></ol> <i>Then <s> is called a <b>subgroup</b> of <g,>. </i> <br /><i>For any group <g,>, <</i>{<i>e</i>}<i>, *> and <g,> are trivial subgroups. All other subgroups are called <b>proper subgroups</b>. </i><br /> <br /><b style="">Examples: </b><br /> <ol><li> (Q , +) is a subgroup of (R, +).</li><li> (D<sub>n </sub>, <span style="font-family:Symbol;">à</span>) is a subgroup of (S<sub>n</sub> , <span style="font-family:Symbol;">à</span>).</li><li> ({[0], [2], [4]} , +) and ({[0], [3]} , +) are subgroups of (Z<sub>6</sub> , +).</li><li> Let a be an element of a group <g,>, then G must contain all the integral powers of a, a<sup>r </sup> G, for r Z, thus, the cyclic group generated by an element ‘a’ is a subgroup of <g,>.</li></ol> <br /><b>Theorem 3.2-1:</b><br /> <br />A non-empty subset S of G is a subgroup of <g,> if and only if for any pair of elements a, b <span style="font-family:Symbol;">Î</span> S, a * b<sup>-1 </sup> <span style="font-family:Symbol;">Î </span>S.<br /> <br /><b>Proof:</b><br /> <br />Assume that S is a subgroup. For any pair a, b <span style="font-family:Symbol;">Î</span> S, we have a, b<sup>-1</sup> <span style="font-family:Symbol;">Î</span> S, thus a * b<sup>-1</sup> <span style="font-family:Symbol;">Î</span> S.<br /> <br />To prove the converse, let us assume that for any pair a, b <span style="font-family:Symbol;">Î </span>S, a * b<sup>-1 </sup><span style="font-family:Symbol;">Î </span>S.<br /> Since S is nonempty, let a <span style="font-family:Symbol;">Î </span>S. Then by taking b = a , we have a * a<sup>-1</sup> = e <span style="font-family:Symbol;">Î </span>S.<br /> For every a <span style="font-family:Symbol;">Î</span> S, consider the pair a, e in S, then we have, e*a<sup>-1</sup> <span style="font-family:Symbol;">Î</span> S, that is, a<sup>-1</sup> <span style="font-family:Symbol;">Î</span> S.<br /> Finally, for any a, b <span style="font-family:Symbol;">Î</span> S, consider the pair a, b<sup>-1</sup><span style="font-family:Symbol;">Î</span> S, then we have a*(b<sup>-1</sup>)<sup>-1</sup> = a * b <span style="font-family:Symbol;">Î</span> S.<br /> Hence <s> is a subgroup of <g>.<br /> <br /><i>Let <g,> and <h, face="Symbol">D</span>> be two groups. A mapping g : G <span style="font-family:Symbol;">®</span> H is called a <b>group homomorphism</b> from <g,> to<br /> <h, face="Symbol">D</span>>, if for any a, b <span style="font-family:Symbol;">Î</span> <i>G,</i></i> g(a * b) = g(a) <span style="font-family:Symbol;">D</span> g(b).<br /> <br /><b>Examples:</b><br /><ol><li> Let G = (Z , +) and let H = ({1, -1} , •), where • is the usual multiplication. <br />Define f : G <span style="font-family:Symbol;">®</span> H by <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.2/Image/Sectio1.gif" shapes="_x0000_i1025" width="140" align="absmiddle" height="48" /> <br />Then, f(m + n) = f(m) • f(n), for m, n <span style="font-family:Symbol;">Î</span> Z.<br /> Thus, f is a homomorphism.</li><li> Let G = (Z , +) and let H = (Z<sub>n</sub> ,+). <br />Define f : G <span style="font-family:Symbol;">®</span> H by f(a) = [a (mod n)].<br /> Then, f(a+b) = [(a+b)(mod n)]<br /> = [a(mod n)] + [b(mod n)] <br /> = f(a) + f(b).<br /> Thus, f is a homomorphism. </li></ol> <br /><br /> <b>Property 1: </b><i>Group homomorphism preserves the identity element.</i><br /> <br />Let e<sub>G</sub> and e<sub>H</sub> are the identities of the groups <g> and <h, face="Symbol">D</span>> respectively. <br />Let g : G <span style="font-family:Symbol;">®</span> H be a group homomorphism.<br /> <br />Now g(e<sub>G</sub> * e<sub>G</sub>) = g (e<sub>G</sub>) = g (e<sub>G</sub>) <span style="font-family:Symbol;">D</span> g(e<sub>G</sub>), that is, g(e<sub>G</sub>) is idempotant.<br /> <br />But the only idempotant element of a group is its identity, therefore, g(e<sub>G</sub>) = e<sub>H</sub>.<br /> <br /><b>Property 2: </b><i>Group homomorphism preserves inverses.</i><br /> <br />Let a <span style="font-family:Symbol;">Î</span> G and a<sup>-1</sup> be its inverse.<br /> <br />Then, g(a*a<sup>-1</sup>) = g (e<sub>G</sub>) = e<sub>H</sub> = g (a) <span style="font-family:Symbol;">D</span> g (a<sup>-1</sup>)<br /> <br />Similarly, g(a<sup>-1</sup> *a ) = g(e<sub>G</sub>) = e<sub>H</sub> = g(a<sup>-1</sup>) <span style="font-family:Symbol;">D</span> g(a).<br /> <br />Therefore, g(a<sup>-1</sup>) is the inverse of g(a) in <h, face="Symbol">D</span>>.<br /> <br /><b>Property 3: </b><i>Group homomorphism preserves subgroups.</i><br /> <br />Let g : <g> <span style="font-family:Symbol;">®</span> <h face="Symbol">D</span>> be a group homomorphism.<br /> <br />Let <s,> be a subgroup of <g,> .<br /> <br />Therefore e<sub>H</sub> = g(e<sub>G</sub>) <span style="font-family:Symbol;">Î</span> g(S).<br /> <br />For any a <span style="font-family:Symbol;">Î</span> S, a<sup>-1</sup> <span style="font-family:Symbol;">Î</span> S and g(a) as well as g(a<sup>-1</sup>) = [g (a)]<sup>-1</sup> are in g (S).<br /> <br />Also for a, b <span style="font-family:Symbol;">Î</span> S, a * b <span style="font-family:Symbol;">Î</span> S and hence<br /> <br />g(a), g(b) <span style="font-family:Symbol;">Î</span> g(S) and g(a*b) <span style="font-family:Symbol;">Î</span> g(S).<br /> <br />Since g is a homomorphism, we have, g(a) <span style="font-family:Symbol;">D</span> g(b) = g(a * b) <span style="font-family:Symbol;">Î</span> g(S).<br /> <br />Thus, <g(s), face="Symbol">D</span>> is a subgroup of <h face="Symbol">D</span>>.<br /> <br /><br /> <br /><i>A group homomorphism g is called <b>monomorphism </b>or <b>epimorphism</b> or <b> isomorphism </b>depending on whether g is<br /> "one-one" or "onto" or "one-one and onto" respectively.</i><br /> <br /><i>A homomorphism from a group (G , *) to (G , *) is called <b>endomorphism</b> while an isomorphism of (G, *) into (G, *)<br /> is called <b>automorphism</b>.</i> <br /><i>Let g be a group homomorphism from <g,> to <h, face="Symbol">D</span>>. The set of element of G which are mapped onto e<sub>H</sub>, the<br /> identity of H, is called the <b>kernel of the homomorphism</b> and it is denoted by <b>ker(g)</b>.</i><br /> <br /> <br /> <br /><b>Theorem 3.2-2:</b> <br /> <br />The kernel of a homomorphism g from a group <g,> to <h, face="Symbol">D</span>> is a subgroup of <g,>.<br /> <br /><b>Proof:</b> <br />Let g : <g> <span style="font-family:Symbol;">®</span> <h face="Symbol">D</span>> be a homomorphism.<br /> <br />Since g(e<sub>G</sub>) = e<sub>H</sub>, e<sub>G</sub> <span style="font-family:Symbol;">Î</span> ker (g)<br /> <br />Also, if a, b <span style="font-family:Symbol;">Î</span> ker(g), then, g(a) = g(b) = e<sub>H</sub>.<br /> <br />As g(a * b) = g(a) <span style="font-family:Symbol;">D</span> g(b) = e<sub>H </sub><span style="font-family:Symbol;">D</span> e<sub>H</sub> = e<sub>H</sub>, we have a * b <span style="font-family:Symbol;">Î</span> ker(g).<br /> <br />Finally, if a <span style="font-family:Symbol;">Î</span> ker (g), then g(a<sup>-1</sup>) = [g(a)]<sup>-1</sup> = e<sub>H</sub><sup>-1</sup> = e<sub>H</sub>.<br /> <br />Thus, for a <span style="font-family:Symbol;">Î</span> ker(g), a<sup>-1</sup> <span style="font-family:Symbol;">Î</span> ker (g).<br /> <br />Hence ker (g) is a subgroup of (G, *).<br /> <br /><br /> <br /><b>Theorem 3.2-3: </b><br /> <br />Any infinite cyclic group is isomorphic to Z.<br /> <br /><b>Proof:</b><br /> <br />Let G = (a) be an infinite cyclic group.<br /> <br />Define g : <z,> <span style="font-family:Symbol;">®</span> <g,> by g(n) = a<sup>n</sup>.<br /> <br />Then, g(m+n) = a<sup>m+n</sup> = a<sup>m</sup> * a<sup>n</sup> = g(m) * g(n).<br /> <br />Therefore, g is a homomorphism.<br /> <br />As <g> is an infinite group , if m <span style="font-family:Symbol;">¹</span> n then a<sup>m</sup> <span style="font-family:Symbol;">¹</span> a<sup>n</sup> .<br /> <br />Hence, if m <span style="font-family:Symbol;">¹</span> n, g(m) <span style="font-family:Symbol;">¹</span> g(n) .<br /> <br />Thus, g is one-one.<br /> <br />For every b <span style="font-family:Symbol;">Î</span> G, there exists some m <span style="font-family:Symbol;">Î</span> Z such that b = a<sup>m</sup> .<br /> <br />Thus, g(m) = a<sup>m</sup> = b.<br /> <br />Hence, g is onto. <br /> <br /> <br /> <br /><b>Exercise:</b> <br /> <br />Prove that any cyclic group of order n is isomorphic to Z<sub>n</sub>. <br /> <br /><br /> <br /><b>Theorem 3.2-4:</b><br /> Every finite group of order n is isomorphic to a permutation group of degree n.<br /> <br /><b>Proof: </b><br /> <br />Let <g,> be a group of order n. We know that every row and column in the composition table of <g,> represent a<br /> permutation of the element of G. Corresponding to an element a <span style="font-family:Symbol;">Î</span> G we denote p<sub>a</sub>, the permutation given by the column under<br /> ‘a’ in the composition table.<br /> <br />Thus, p<sub>a</sub>(c) = c * a, for any c <span style="font-family:Symbol;">Î</span> G.<br /> <br />Let the set of permutation be denoted by <b>P</b>.Then <b>P </b>has n elements. We shall show that <<b>P</b>, <span style="font-family:Symbol;">à</span>> is a group where <span style="font-family:Symbol;">à</span> denotes the<br /> right composition of the permutation of <b>P</b>.<br /> <br />Since e <span style="font-family:Symbol;">Î</span> G, p<sub>e</sub> <span style="font-family:Symbol;">Î</span> <b>P</b> and p<sub>e</sub><span style="font-family:Symbol;">à</span> p<sub>a</sub> = p<sub>a</sub> <span style="font-family:Symbol;">à</span> p<sub>e</sub> = p<sub>a</sub>, for any a <span style="font-family:Symbol;">Î</span> G.<br /> <br />Also, for any a <span style="font-family:Symbol;">Î</span> G, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.2/Image/painv.gif" width="25" align="absmiddle" border="0" height="21" /> <span style="font-family:Symbol;">à</span> p<sub>a</sub> = p<sub>e</sub>.<br /> <br />Further, for any a, b <span style="font-family:Symbol;">Î</span> G, p<sub>a</sub> <span style="font-family:Symbol;">à</span> p<sub>b</sub> = p <sub>a*b</sub> ………… (1)<br /> <br />The equation (1) follows from the fact that<br /> <br />(p<sub>a</sub> <span style="font-family:Symbol;">à</span> p<sub>b</sub>)(c) = (c * a) * b (by right composition)<br /> = c * (a *b) (by associativity)<br /> = p<sub>a*b</sub>(c)<br /> <br />Thus (<b>P </b>, <span style="font-family:Symbol;">à</span>) is a group.<br /> <br />Consider the mapping f : G <span style="font-family:Symbol;">®</span> <b>P</b> given by f(a) = p<sub>a</sub> for any a <span style="font-family:Symbol;">Î</span> G.<br /> <br />Then, f(a * b) = p<sub>a*b</sub> = p<sub>a</sub> <span style="font-family:Symbol;">à</span> p<sub>b</sub> = f(a) <span style="font-family:Symbol;">à</span> f(b).<br /> <br />Thus f is a homomorphism.<br /> <br />If a <span style="font-family:Symbol;">¹</span> b then p<sub>a</sub> <span style="font-family:Symbol;">¹</span> p<sub>b</sub>, that is, f(a) <span style="font-family:Symbol;">¹</span> f(b).<br /> <br />Hence f is one-one.<br /> <br />Further, for every p<sub>a</sub><span style="font-family:Symbol;">Î</span> <b>P</b>, there exists a <span style="font-family:Symbol;">Î</span> G such that f(a) = p<sub>a</sub>.<br /> <br />Thus f is onto.<br /> <br />Hence f is an isomorphism.<br /> <br />Hence the theorem.Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-53615879249358367302008-08-12T04:24:00.002-07:002008-12-23T01:55:02.555-08:00GroupsThe theory of group arose from the theory of equations, later on the group of transformations was extended and generalized as<br /> an abstract group. Group theory has played a major role in finding new connections between various branches of mathematics<br /> especially in the studies of geometric problems. Group theory has tremendous applications in physics and new types of<br /> interesting and significant applications of group theory are being explored in biological sciences and theoretical computer<br /> science too.<br /> <br />A group (G, *) is a non-empty set G, together with a binary operation * satisfying the following axioms.<br />For all a, b, c <span style="font-family:Symbol;">Î</span> G,<br /> <ol><li> a * (b * c)=(a * b) * c.</li><li> There exists an element e in G such that a * e = a = e * a.</li><li> For each a <span style="font-family:Symbol;">Î</span> G, there exists an element a<span style="font-family:Symbol;">¢</span> <span style="font-family:Symbol;">Î</span> G such that a * a<span style="font-family:Symbol;">¢</span> = e = a<span style="font-family:Symbol;">¢</span> * a.</li></ol> <b> <br />Examples:<br /> </b> <ol><li> (Z, +), (Q, +), (R, +) and (C, +) are all groups, where Z is the set of integers, Q is the set of rational numbers, R is the<br /> set of real numbers and C is the set of complex numbers, and ‘+’ is the usual addition.</li><li> (Z\{0}, ·), where <span style="font-family:UniversalMath1 BT;">.</span> is the usual multiplication, is not a group, since inverse does not exist for every element of Z\{0}.</li><li> (Q\{0}, ·), (R\{0}, ·), (C\{0}, ·) are all groups, where <span style="font-family:UniversalMath1 BT;">.</span> is the usual multiplication.</li><li> Let M<span style="font-size:100%;"><sub>n</sub></span> denote the set of all non-singular matrices over R, then (M<span style="font-size:100%;"><sub>n</sub></span><sub> </sub>,*) is a group, where * is the matrix multiplication<br /> (Prove !).</li><li> Let S be any non-empty set. Then the set of all bijective mappings from S into itself together with the binary operation the<br /> composition of mappings form a group.</li><li> Let Z<span style="font-size:100%;"><sub>m</sub></span> be the set of distinct equivalence classes of integer modulo m, then (Z<span style="font-size:100%;"><sub>m</sub></span>, +) is a group, where<br /> [a] + [b] = [(a + b)(mod m)].</li></ol> <br />From the definition of group it is clear that group always has identity element and each element in a group has inverse. In fact we<br /> can show that identity element in a group is unique and inverse for an element in a group is unique.<br /> <br />Suppose a group has two identity elements, say e and e<span style="font-family:Symbol;">¢</span>, then by taking an element a = e<span style="font-family:Symbol;">¢</span> in axiom(2) of group we have,<br /> e<span style="font-family:Symbol;">¢</span> * e = e<span style="font-family:Symbol;">¢</span> = e * e<span style="font-family:Symbol;">¢</span> ……… <span style="font-family:Times New Roman;">( i )</span><br /> <br />Similarly by taking a=e in axiom(2) of the group, we have<br /> e * e<span style="font-family:Symbol;">¢</span> = e = e<span style="font-family:Symbol;">¢</span> * e ……… <span style="font-family:Times New Roman;">( ii )</span><br /> <br />Thus, we have from <span style="font-family:Times New Roman;">( i )</span> and <span style="font-family:Times New Roman;">( ii )</span> ,<br /> e<span style="font-family:Symbol;">¢</span> = e * e<span style="font-family:Symbol;">¢</span> = e.<br /> <br />Thus, in a group identity element is unique.<br /> <br />Similarly, if an element ‘a’ has two inverses, say a<span style="font-family:Symbol;">¢</span> and a<span style="font-family:Symbol;">¢¢</span>,<br /> then we have a * a<span style="font-family:Symbol;">¢</span> = e = a<span style="font-family:Symbol;">¢</span> * a and a * a<span style="font-family:Symbol;">¢¢</span> = e = a<span style="font-family:Symbol;">¢¢</span> * a.<br /> Now, a<span style="font-family:Symbol;">¢</span> = a<span style="font-family:Symbol;">¢</span> * e<br /> = a<span style="font-family:Symbol;">¢</span> * (a * a<span style="font-family:Symbol;">¢¢</span> )<br /> = (a<span style="font-family:Symbol;">¢</span> * a) * a<span style="font-family:Symbol;">¢¢</span><br /> = e * a<span style="font-family:Symbol;">¢¢</span><br /> = a<span style="font-family:Symbol;">¢¢</span><br /> <br />Thus, in a group inverse of an element is unique.<br /> <i> <br />A group (G ,*) is called a <b>finite group</b> if the cardinality of G is finite. The number of elements in a finite group,<br /> i.e. |G| , is called the <b>order of the group</b> and is denoted by the symbol <b>o(G)</b>. A group which is not finite is called an <b><br /> infinite group.</b><br /> </i> <br />If a group is finite, then definition of its binary operation can be expressed in the form of a table. For example, let<br /> G = {e, a, b, c}, then the binary operation * on G can be expressed by the following table:<br /> <br /><b> Table 1<br /> </b> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image52.gif" width="106" height="117" /><br /><br />Observe that the element x * y obtained by operating the elements x and y by using the operation * is the element in the row<br /> containing x and the column containing y. For instance, a * b = c, a * a = e, etc.<br /> <i> <br />A group (G, * ) is called an <b>abelian group</b> if a * b = b * a, for all a, b <span style="font-family:Symbol;">Î</span> G.<br /> <br />Abelian groups are sometimes called <b>commutative groups.<br /> </b></i> <br />Examples 1, 3, 6 and 7 are commutative groups, while Example 4 is not an abelian group.<br /> <br />In fact it is interesting to observe that all finite groups with order <span style="font-family:Symbol;">£ </span>5 are all commutative.<br /> <br />In the next section we discuss about permutation groups. This is an important example of class of groups that are non-abelian.<br /> <br /><br /> <p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Section3.1.htm#3.1%20Introduction">Back to top</a></span> </p><p style="line-height: 150%;" align="left"> <b> <span style="font-size:130%;"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.5/Image/Sectio2.gif" width="767" border="0" height="10" /><br /> </span> <nobr><br /><a name="3.1.1 Permutation Groups">3.1.1 Permutation Groups</a><br /> </nobr> </b> <nobr> <br />In this subsection we discuss about a popular class of groups called <i>permutation groups. </i>In particular, we discuss about two<br /> interesting permutation groups, the symmetric groups (S<sub>n</sub>, <span style="font-family:Symbol;">à</span>) and the dihedral group (D<sub>n</sub>, <span style="font-family:Symbol;">à</span>). The permutation groups play an<br /> important role in the studies of finite combinatorial and geometrical structures.<br /> <b> <br /><a name="3.1.1.1 Symmetric Groups">3.1.1.1 Symmetric Groups</a><br /> </b><i> <br />Any one-to-one mapping of a set S onto S is called <b>permutation of S.<br /> </b> </i> <br />Let S = {a, b, c} be a set and p denote a permutation of elements of S, that is, p : S <span style="font-family:Symbol;">® </span>S , is a bijective mapping. <br />Suppose p(a) = c, p(b) = a and p(c) = b, then the classical way of representing p is<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image53.gif" width="97" align="absmiddle" height="48" /> , where the image of any element of S is entered below the element.<br /> <br />Consider now a set S={a, b} consisting of only two elements and let the permutations on the elements of this set be denoted<br /> by p<sub>1</sub> and p<sub>2</sub>, where<br /> <br /> <sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image54.gif" width="192" height="48" />.</sub><br /> <br />It is clear that p<sub>1</sub> and p<sub>2</sub> are the only possible permutations on the set S.<br /> <br />Let <span style="font-family:Symbol;">à</span> denote a binary operation on S representing the (right) composition of permutation. By p<sub>i </sub><span style="font-family:Symbol;">à </span>p<sub>j</sub> ,<br />for i, j =1, 2, we mean the permutation obtained by permuting the elements of S by an application of p<sub>i</sub> followed by an<br /> application of the permutation p<sub>j.</sub><br /> <br />For example, p<sub>i</sub> <span style="font-family:Symbol;">à</span> p<sub>j</sub> (a) = p<sub>j </sub>( p<sub>i</sub>(a) ).<br /> <br />That is, in the above example, we have p<sub>2 </sub> <span style="font-family:Symbol;">à</span> p<sub>1</sub>(a) = p<sub>1</sub>( p<sub>2</sub>(a)) = p<sub>1</sub>(b) = b.<br /> p<sub>2 </sub> <span style="font-family:Symbol;">à</span> p<sub>1</sub>(b) = p<sub>1</sub>(p<sub>2</sub>(b)) = p<sub>1</sub>(a) = a.<br /> Thus p<sub>2 </sub> <span style="font-family:Symbol;">à</span> p<sub>1</sub> = p<sub>2</sub> .<br /> <br />So, we have the following composition table.<br /> <br /><b> Table 2<br /> </b> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image55.gif" width="84" height="72" /><br /> <br /> <br />It follows from the above table that (S<sub>2</sub>, <span style="font-family:Symbol;">à</span>) is a group of order 2.<br /> <br />Observe that (S<sub>2</sub>, <span style="font-family:Symbol;">à</span>) is independent of the elements of the set S (= {a, b}) but depends upon the number of elements in the set.<br /> Any other set of two elements will generate the same permutation group (S<sub>2</sub>, <span style="font-family:Symbol;">à</span>).<br /> <br />Next consider 3!(= 6) permutations of the elements of the set {1, 2, 3}. Let us denote the set of all permutations by<br /> S<sub>3</sub> = {p<sub>1</sub>, p<sub>2</sub>, …, p<sub>6</sub>}, where<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image56.gif" width="104" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image57.gif" width="105" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image58.gif" width="105" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image59.gif" width="105" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image60.gif" width="105" align="absmiddle" height="48" /> and<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image61.gif" width="105" align="absmiddle" height="48" />.<br /> <br />The following composition table shows that (S<sub>3</sub>, <span style="font-family:Symbol;">à</span>) is a group.<br /> <br /><b> Table 3</b> : <b>Composition Table of (S<sub>3</sub> , <span style="font-family:Symbol;">à</span> )</b><br /> <table align="center" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top"> <br /><span style="font-family:Symbol;">à</span></td> <td valign="top"><b> <br />p<sub>1</sub></b></td> <td valign="top"><b> <br />p<sub>2</sub></b></td> <td valign="top"><b> <br />p<sub>3</sub></b></td> <td valign="top"><b> <br />p<sub>4</sub></b></td> <td valign="top"><b> <br />p<sub>5</sub></b></td> <td valign="top"><b> <br />p<sub>6</sub></b></td> </tr> <tr> <td valign="top"><b> <br />p<sub>1</sub></b></td> <td valign="top"> <br />p<sub>1</sub></td> <td valign="top"> <br />p<sub>2</sub></td> <td valign="top"> <br />p<sub>3</sub></td> <td valign="top"> <br />p<sub>4</sub></td> <td valign="top"> <br />p<sub>5</sub></td> <td valign="top"> <br />p<sub>6</sub></td> </tr> <tr> <td valign="top"><b> <br />p<sub>2</sub></b></td> <td valign="top"> <br />p<sub>2</sub></td> <td valign="top"> <br />p<sub>1</sub></td> <td valign="top"> <br />p<sub>5</sub></td> <td valign="top"> <br />p<sub>6</sub></td> <td valign="top"> <br />p<sub>3</sub></td> <td valign="top"> <br />p<sub>4</sub></td> </tr> <tr> <td valign="top"><b> <br />p<sub>3</sub></b></td> <td valign="top"> <br />p<sub>3</sub></td> <td valign="top"> <br />p<sub>6</sub></td> <td valign="top"> <br />p<sub>1</sub></td> <td valign="top"> <br />p<sub>5</sub></td> <td valign="top"> <br />p<sub>4</sub></td> <td valign="top"> <br />p<sub>2</sub></td> </tr> <tr> <td valign="top"><b> <br />p<sub>4</sub></b></td> <td valign="top"> <br />p<sub>4</sub></td> <td valign="top"> <br />p<sub>5</sub></td> <td valign="top"> <br />p<sub>6</sub></td> <td valign="top"> <br />p<sub>1</sub></td> <td valign="top"> <br />p<sub>2</sub></td> <td valign="top"> <br />p<sub>3</sub></td> </tr> <tr> <td valign="top"><b> <br />p<sub>5</sub></b></td> <td valign="top"> <br />p<sub>5</sub></td> <td valign="top"> <br />p<sub>4</sub></td> <td valign="top"> <br />p<sub>2</sub></td> <td valign="top"> <br />p<sub>3</sub></td> <td valign="top"> <br />p<sub>6</sub></td> <td valign="top"> <br />p<sub>1</sub></td> </tr> <tr> <td valign="top"><b> <br />p<sub>6</sub></b></td> <td valign="top"> <br />p<sub>6</sub></td> <td valign="top"> <br />p<sub>3</sub></td> <td valign="top"> <br />p<sub>4</sub></td> <td valign="top"> <br />p<sub>2</sub></td> <td valign="top"> <br />p<sub>1</sub></td> <td valign="top"> <br />p<sub>5</sub></td> </tr> </tbody></table> <br /><br /> <br />It is clear from the table that (S<sub>3</sub>, <span style="font-family:Symbol;">à</span>) is not abelian. In fact S<sub>3</sub> is the smallest non-abelian group.<br /> <i> <br />In general the set S<span style="font-size:100%;"><sub>n</sub></span> of all permutations of n elements is a permutation group </i>(S<span style="font-size:100%;"><sub>n</sub></span>, <span style="font-family:Symbol;">à</span>)<i>, also called <b>symmetric group</b></i>.<br /> <br />It is interesting to observe that the symmetric groups (S<sub>n</sub>, <span style="font-family:Symbol;">à</span>), n <span style="font-family:Symbol;">³ </span>3, are all non-abelian.<br /> <br />In the permutation group, the number elements on which the permutations are defined is called the <b><i>degree of the permutation<br /> group<br /> </i></b> <br />Thus in the symmetric group S<sub>n</sub>, degree is n and the order is n!.<br /> <br /><br /></nobr> </p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Section3.1.htm#3.1%20Introduction">Back to top</a></span> </p><p style="line-height: 150%;" align="left"> <b><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.5/Image/Sectio2.gif" width="767" border="0" height="10" /></b><br /> <nobr><b> <br /><a name="3.1.1.2 Dihedral Group">3.1.1.2 Dihedral Group</a><br /> </b> <br />By considering the symmetries of regular polygon we obtain another interesting permutation groups called <b><i>dihedral groups</i></b>.<br /> <br />First we discuss the permutation group obtained from an equilateral triangle.<br /> <br />Consider an equilateral triangle with vertices 1, 2 and 3 as shown in Figure 3.1. Consider the following transformation of the<br /> triangle itself<br /> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Sectio1.gif" width="240" border="0" height="184" /><br /><br /> <b><br /> <br /> <br /> <br /> <br /> Figure 3.1</b><br /> <ol><li> Three rotations through the center through 0<span style="font-family:Symbol;">°</span> , 120<span style="font-family:Symbol;">°</span> , 240<span style="font-family:Symbol;">°</span> (in counter clockwise).</li><li> Three reflections along the three bisectors. We call any one of the above six transformations as symmetry.</li></ol> <br />On the six transformations (the symmetries), we consider the composition of transformation as a binary operation. The<br /> symmetries form a group which has its identity element, the identity transformation (corresponding to 0<span style="font-family:Symbol;">°</span> rotation). Rotations<br /> through 120<span style="font-family:Symbol;">°</span> and through 240<span style="font-family:Symbol;">°</span> are mutual inverse of each other. For every other reflection the inverse is itself.<br /> <br />Thus the symmetries form a group.<br /> <br />The product (composition) of rotation through 120<sup>o</sup> with reflection along the bisector through vertex 1 is the reflection along the<br /> bisector through 2. But the product in the reverse order, that is, the product of reflection along the bisector through vertex 1 and<br /> the rotation through 120<span style="font-family:Symbol;">°</span> is the reflection along the bisector through 3. Hence, this group is non-abelian.<br /> <br />The above example can also be interpreted as follows:<br /> <br />Each symmetry permutes the vertices of the triangle and each permutation of the vertices of the triangle defines a symmetry. The<br /> three rotations correspond respectively, to the permutations<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image62.gif" width="346" height="48" /><br /> <br />and the three reflections along the bisectors through 1, 2 and 3 respectively corresponding to the permutations<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image63.gif" width="348" height="48" />.<br /> <br />The product of the symmetries corresponds to the product of the permutations. Thus, the six permutations form a group. This<br /> group is denoted by D<sub>3</sub>. It is very clear that D<sub>3 </sub>coincides with S<sub>3</sub>, that is, S<sub>3 </sub>and D<sub>3 </sub>are one and the same.<br /> <br />Now we consider the symmetries of the square. Consider a square with vertices 1, 2, 3 and 4.<br /> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Sectio2.gif" width="196" border="0" height="188" /><br /> <br />Consider the following transformation of square into itself.<br /> <ol type="i"><li> The four rotation about the center through 90<span style="font-family:Symbol;">°</span>, 180<span style="font-family:Symbol;">°</span>, 270<span style="font-family:Symbol;">°</span>, 0<span style="font-family:Symbol;">°</span> respectively (in the counter clockwise direction) about O.<br /> These four rotations can be interpreted as the following permutation of the vertices</li></ol> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image64.gif" width="122" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image65.gif" width="124" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image66.gif" width="124" align="absmiddle" height="48" /> and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image67.gif" width="124" align="absmiddle" height="48" />.<br /><ol start="2" type="i"><li> The two reflections about the lines AA<span style="font-family:Symbol;">¢</span> and BB<span style="font-family:Symbol;">¢</span> and two more reflections about the diagonals 13 and 24. These reflections<br /> can be interpreted as the following permutations of the vertices</li></ol> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image68.gif" width="124" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image69.gif" width="124" align="absmiddle" height="48" />, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image70.gif" width="122" align="absmiddle" height="48" /> and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image71.gif" width="124" align="absmiddle" height="48" />.<br /> <br />Let D<sub>4</sub> = {r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub>,…, r<sub>8</sub>}. The compositions of the rotations and the reflections of D<sub>4</sub> is given in the table.<br /> <br /> <b> Table 4:</b> <b>Composition table of (D<sub>4</sub> , <span style="font-family:Symbol;">à</span> ) </b><br /> <table width="394" align="center" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="8%"> <br /><span style="font-family:Symbol;">à</span></td> <td valign="top" width="12%"><b> <br />r<sub>1</sub></b></td> <td valign="top" width="12%"><b> <br />r<sub>2</sub></b></td> <td valign="top" width="12%"><b> <br />r<sub>3</sub></b></td> <td valign="top" width="12%"><b> <br />r<sub>4</sub></b></td> <td valign="top" width="12%"><b> <br />r<sub>5</sub></b></td> <td valign="top" width="10%"><b> <br />r<sub>6</sub></b></td> <td valign="top" width="10%"><b> <br />r<sub>7</sub></b></td> <td valign="top" width="10%"><b> <br />r<sub>8</sub></b></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>1</sub></b></td> <td valign="top" width="12%"> <br />r<sub>2</sub></td> <td valign="top" width="12%"> <br />r<sub>3</sub></td> <td valign="top" width="12%"> <br />r<sub>7</sub></td> <td valign="top" width="12%"> <br />r<sub>1</sub></td> <td valign="top" width="12%"> <br />r<sub>8</sub></td> <td valign="top" width="10%"> <br />r<sub>7</sub></td> <td valign="top" width="10%"> <br />r<sub>5</sub></td> <td valign="top" width="10%"> <br />r<sub>6</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>2</sub></b></td> <td valign="top" width="12%"> <br />r<sub>3</sub></td> <td valign="top" width="12%"> <br />r<sub>4</sub></td> <td valign="top" width="12%"> <br />r<sub>1</sub></td> <td valign="top" width="12%"> <br />r<sub>2</sub></td> <td valign="top" width="12%"> <br />r<sub>6</sub></td> <td valign="top" width="10%"> <br />r<sub>5</sub></td> <td valign="top" width="10%"> <br />r<sub>8</sub></td> <td valign="top" width="10%"> <br />r<sub>7</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>3</sub></b></td> <td valign="top" width="12%"> <br />r<sub>4</sub></td> <td valign="top" width="12%"> <br />r<sub>1</sub></td> <td valign="top" width="12%"> <br />r<sub>2</sub></td> <td valign="top" width="12%"> <br />r<sub>3</sub></td> <td valign="top" width="12%"> <br />r<sub>7</sub></td> <td valign="top" width="10%"> <br />r<sub>8</sub></td> <td valign="top" width="10%"> <br />r<sub>6</sub></td> <td valign="top" width="10%"> <br />r<sub>5</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>4</sub></b></td> <td valign="top" width="12%"> <br />r<sub>1</sub></td> <td valign="top" width="12%"> <br />r<sub>2</sub></td> <td valign="top" width="12%"> <br />r<sub>3</sub></td> <td valign="top" width="12%"> <br />r<sub>4</sub></td> <td valign="top" width="12%"> <br />r<sub>5</sub></td> <td valign="top" width="10%"> <br />r<sub>6</sub></td> <td valign="top" width="10%"> <br />r<sub>7</sub></td> <td valign="top" width="10%"> <br />r<sub>8</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>5</sub></b></td> <td valign="top" width="12%"> <br />r<sub>7</sub></td> <td valign="top" width="12%"> <br />r<sub>6</sub></td> <td valign="top" width="12%"> <br />r<sub>8</sub></td> <td valign="top" width="12%"> <br />r<sub>5</sub></td> <td valign="top" width="12%"> <br />r<sub>4</sub></td> <td valign="top" width="10%"> <br />r<sub>2</sub></td> <td valign="top" width="10%"> <br />r<sub>1</sub></td> <td valign="top" width="10%"> <br />r<sub>3</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>6</sub></b></td> <td valign="top" width="12%"> <br />r<sub>8</sub></td> <td valign="top" width="12%"> <br />r<sub>5</sub></td> <td valign="top" width="12%"> <br />r<sub>7</sub></td> <td valign="top" width="12%"> <br />r<sub>6</sub></td> <td valign="top" width="12%"> <br />r<sub>2</sub></td> <td valign="top" width="10%"> <br />r<sub>4</sub></td> <td valign="top" width="10%"> <br />r<sub>3</sub></td> <td valign="top" width="10%"> <br />r<sub>1</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>7</sub></b></td> <td valign="top" width="12%"> <br />r<sub>6</sub></td> <td valign="top" width="12%"> <br />r<sub>8</sub></td> <td valign="top" width="12%"> <br />r<sub>5</sub></td> <td valign="top" width="12%"> <br />r<sub>7</sub></td> <td valign="top" width="12%"> <br />r<sub>3</sub></td> <td valign="top" width="10%"> <br />r<sub>1</sub></td> <td valign="top" width="10%"> <br />r<sub>4</sub></td> <td valign="top" width="10%"> <br />r<sub>2</sub></td> </tr> <tr> <td valign="top" width="8%"><b> <br />r<sub>8</sub></b></td> <td valign="top" width="12%"> <br />r<sub>5</sub></td> <td valign="top" width="12%"> <br />r<sub>7</sub></td> <td valign="top" width="12%"> <br />r<sub>6</sub></td> <td valign="top" width="12%"> <br />r<sub>8</sub></td> <td valign="top" width="12%"> <br />r<sub>1</sub></td> <td valign="top" width="10%"> <br />r<sub>3</sub></td> <td valign="top" width="10%"> <br />r<sub>2</sub></td> <td valign="top" width="10%"> <br />r<sub>4</sub></td> </tr> </tbody></table> <br /><br /> <br />Thus, from the above table it is clear that (D<sub>4</sub>, <span style="font-family:Symbol;">à</span> ) is a permutation group of order 8 and degree 4 and it is called as <i><b> dihedral<br /> group of degree 4</b></i>. Also, from the table it is clear that D<sub>4</sub> is not abelian.<br /> <br />In general, the set of all rigid rotations of a regular polygon of n sides under the composition <span style="font-family:Symbol;">à</span> is a group (D<sub>n</sub>, <span style="font-family:Symbol;">à</span>), where D<sub>n</sub> is<br /> of order 2n and degree n<sub> </sub>. The group <b> (D<sub>n</sub>, <span style="font-family:Symbol;">à</span> )</b> is called dihedral group of degree n.<br /> <b> <br />Remark:<br /> </b><br />It is evident from the definition of (S<sub>n</sub>, <span style="font-family:Symbol;">à</span>) that (D<sub>n</sub>, <span style="font-family:Symbol;">à</span>) is a subgroup of (S<sub>n</sub>, <span style="font-family:Symbol;">à</span>) and it is non-abelian.<br /> <br /> <b> </b></nobr> </p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Section3.1.htm#3.1%20Introduction">Back to top</a></span> </p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.5/Image/Sectio2.gif" width="767" border="0" height="10" /> <br /> <nobr><a name="3.1.2 Cyclic Groups"><br /> 3.1.2 Cyclic Groups</a><br /> </nobr> </b> <nobr> <br />Here we define and discuss another important class of groups called cyclic groups.<br />Let (G , *) be a group and a <span style="font-family:Symbol;">Î </span>G.<br />Define a<sup>0</sup> = e, a<sup>n+1</sup> = a<sup>n </sup>* a, for n <span style="font-family:Symbol;">Î</span> N. and (a<sup>-1</sup>)<span style="font-size:100%;"><sup>n</sup></span> = a<span style="font-size:100%;"><sup>-n</sup></span>,<span style="font-size:130%;"> </span>for n <span style="font-family:Symbol;">Î</span> N, so that we have defined a<span style="font-size:100%;"><sup>r</sup></span>,<span style="font-size:130%;"> </span>for r <span style="font-family:Symbol;">Î</span> Z, where Z is the set of<br /> integers.<br /><br />Furthermore, a<span style="font-size:100%;"><sup>m</sup></span> * a<span style="font-size:100%;"><sup>n</sup></span> = a<span style="font-size:100%;"><sup>m+n</sup></span>, for m, n <span style="font-family:Symbol;">Î</span> Z.<br /> <br />Let (a) denote the set { a<span style="font-size:100%;"><sup>n</sup></span> / n<span style="font-family:Symbol;">Î</span> Z}.<br /> <i>A group G is called a <b>cyclic group</b> if for some </i> a <span style="font-family:Symbol;">Î </span>G <i> such that </i> G = (a) <i> and the element ‘a’ is called the <b>generator</b> of G.<br /> </i><b> <br />Note :<br /> </b> <br />A cyclic group can have several generators.<br /> <b> <br />Examples:<br /> </b> <ol><li> The group Z of integers is cyclic.<br /> Since Z = (1) = (-1).<br /> Thus, 1 and –1 are the generators of Z.</li><li> The group G = {1, -1, i, -i} with multiplication as its operation is a cyclic group with i and – i as its<br />generators.</li><li> The group G ={ …, <sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image72.gif" width="18" height="20" /></sub>, <sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Image/Image73.gif" width="16" height="18" /></sub>, ¼, ½, 1, 2, 4, 8,16, …} is cyclic with multiplication as its operation with<br />generator ½ .</li><li> The trivial group G = {e} is cyclic with generator e.</li><li> The group (Z<sub>n</sub> , +) of residue classes modulo n is cyclic with generator 1.</li></ol> <b> <br /><br /> Theorem:<br /> </b> <br />If G is a cyclic group then G is abelian.<br /> <b> <br />Proof:<br /> </b> <br />Let G be cyclic. Then G = (a), for some a <span style="font-family:Symbol;">Î </span>G. If x <span style="font-family:Symbol;">Î </span>G and y <span style="font-family:Symbol;">Î </span>G, x = a<sup>m</sup> and y = a<sup>n</sup> for some m, n <span style="font-family:Symbol;">Î </span>Z.<br /> Then xy = a<sup>m</sup>a<sup>n</sup> = a<sup>m+n</sup> = a<sup>n+m</sup> = a<sup>n</sup>a<sup>m</sup> = xy.<br /> Hence, G is abelian.<br /> <b> <br />Theorem:<br /> </b> <br />Let (G , *) be a finite cyclic group generated by an element a <span style="font-family:Symbol;">Î</span> G. If G is of order n,<br /> <br />that is, |G| = n, then, a<sup>n</sup> = e so that G = {a, a<sup>2</sup>, a<sup>3</sup>, …, a<sup>n </sup>= e}. Furthermore, n is the least positive integer for which a<sup>n</sup> = e.<br /> <b> <br />Proof:<br /> </b> <br />Let us assume that for some positive integer m <>m</sup> = e.<br /><br />Since G is a cyclic group, any element of G can be written as a<sup>k</sup>, for some k <span style="font-family:Symbol;">Î </span>Z.<br /> <br />Now by Euclid’s Algorithm, we can write<br /> <br />k = mq + r, where q is some integer and 0 <span style="font-family:Symbol;">£ </span>r <span style="font-family:Symbol;">£ </span>m - 1.<br />This means that a<sup>k</sup> = a<sup>mq + r</sup> = (a<sup>m</sup>)<sup>q</sup> * a<sup>r</sup> = a<sup>r</sup>. <br />That is, every element of G can be expressed as a<sup>r</sup>, for some r, 0 <span style="font-family:Symbol;">£ </span>r <span style="font-family:Symbol;">£ </span>m - 1.<br /> <br />Thus, implying that G has atmost m distinct elements, i.e. |G| = m < n, which is a contradiction.<br /> <br />Hence, a<sup>m </sup>= e, for m < n is not possible.<br /> <b> <br />Claim:</b> <i>The elements a, a<sup>2</sup>, a<sup>3</sup>, …, a<sup>n </sup>are all distinct, where a<sup>n </sup>= e.<br /> </i> <br />Suppose a<sup>i</sup> = a<sup>j</sup>, for i < face="Symbol">£</span> n.<br /> Then a<sup>j - i </sup> = e, where j - i < n, which is again a contradiction.<br /> Hence the claim.<br /> Hence the theorem.<br /> <br />Let (G , *) be a group. <b><i>Order of an element</i></b><i> a <span style="font-family:Symbol;">Î</span> G is the least positive integer m such that a<sup>m</sup> = e.<br /> </i> <br />For example, in the group (Z<sub>7 </sub>\ [0], •), the order of [2] is 3.<br /> <br />For, [2]<sup>1</sup> = [2], [2]<sup>2</sup> = [2]•[2] = [4], [2]<sup>3</sup> = [2]•[2]•[2] = [8] = [1].<br /> <b> <br /><br /> </b></nobr> </p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit3/Section3.1/Section3.1.htm#3.1%20Introduction">Back to top</a></span> </p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.5/Image/Sectio2.gif" width="767" border="0" height="10" /><br /> <nobr><br /><span style="font-size:130%;"><a name="Exercise :">Exercise:</a></span><br /> </nobr></b><nobr> <ol><li> Is it true that (Z<sub>m</sub>\{0), •) is a group, for all m <span style="font-family:Symbol;">Î</span> Z<sup>+</sup> where [a]•[b] = [ab (mod m)]. If the answer is no then what are the<br /> m <span style="font-family:Symbol;">Î</span> Z<sup>+</sup> such that (Z<sub>m</sub>\{0}, •) is a group.</li><li> Prove that in a group (G , *)</li></ol> <blockquote> <ol type="i"><li> ab = ac implies b = c (left cancellation law).</li><li> ba = ca implies b = c (right cancellation law).</li><li> (ab)<sup>-1</sup> = b<sup>-1</sup>a<sup>-1</sup>.</li></ol> </blockquote> <ol start="3"><li> Show that in a group identity element is the only idempotent element.</li><li> Construct the multiplication table for (Z<sub>7</sub> , +) and (Z<sub>7</sub> , • ). If (G , *) is an abelian group, then for all a, b <span style="font-family:Symbol;">Î</span> G show that<br /> (a * b)<sup>n</sup> = a<sup>n</sup> * b<sup>n</sup>.</li><li> Find all those elements in S<sub>3</sub> such that (a * b)<sup>2</sup> <span style="font-family:Symbol;">¹</span> a<sup>2</sup> * b<sup>2</sup>.</li><li> Show that in a group (G , *) if for any a, b <span style="font-family:Symbol;">Î</span> G, (a * b)<sup>2</sup> = a<sup>2</sup> * b<sup>2</sup> , then (G , *) must be abelian . Is the converse true?</li><li> Show that if G is a group such that a<sup>2</sup> = e, for every a <span style="font-family:Symbol;">Î</span> G, then G is abelian.</li><li> Find a solution of the equation ax = b in S<sub>3</sub>, where </li></ol> . <ol start="9"><li> Prove that if G is non-abelian then G is not cyclic.</li></ol></nobr> </p>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-1231804867370134602008-08-12T04:24:00.001-07:002008-12-23T01:55:02.555-08:00Grammars<span style="font-size:100%;"><b><a name="Grammars"></a></b></span><br /> <br />The field of software in the Computer Science is growing rapidly fast due to the development of well-structured high-level languages. Grammars impart a structure to a program in a high-level language that is useful for translation into a low-level language (machine language). The study of grammar constitutes an important area of computer science called Formal languages. This fascinating area emerged in the middle of 1950, due to the effort of Noam Chomsky, who gave an interesting mathematical model of a grammar in connection with the study of natural languages.<br /><br /> <i>A <b>grammar</b> is defined by a 4-tuple G = (V<sub>N </sub>, V<sub>T </sub>, S , </i> <span style="font-family:Symbol;">f</span> <i> ), where V<sub>T</sub> and V<sub>N</sub> are sets of terminal and non-terminal symbols respectively, S a distinguished elements of V<sub>N</sub>, is called the starting symbol, </i> <span style="font-family:Symbol;">f </span><i>is a finite subset of the relation from<br /> <br />(V<sub>T</sub> <span style="font-family:Symbol;">È</span> V<sub>N</sub>)<sup>*</sup>V<sub>N</sub>(V<sub>T</sub> <span style="font-family:Symbol;">È</span> V<sub>N</sub>)<sup>*</sup> to (V<sub>T</sub> <span style="font-family:Symbol;">È</span> V<sub>N</sub>)<sup>*</sup>.<br /> <br />The elements of <span style="font-family:Symbol;">f</span> are called production or a rewriting rule.<br /> </i><b> <br />Example 1:</b><br /> <br />G<sub>1</sub> = ({E, T, F} , {a, +, *, (, )} , E, <span style="font-family:Symbol;">f</span> ) , where <span style="font-family:Symbol;">f</span> contains the productions<br /> <br />E <span style="font-family:Symbol;">®</span> E + T, E <span style="font-family:Symbol;">®</span> T, T <span style="font-family:Symbol;">®</span> T * F, F <span style="font-family:Symbol;">®</span> (E), F<span style="font-family:Symbol;">®</span> a,<br /> This grammar is used for generating a certain set of arithmetic expression.<br /> <b> <br />Example 2:<br /> </b> <br />G<sub>2</sub> = ({S, B, C} , {a, b, c} , S , <span style="font-family:Symbol;">f</span> ), where <span style="font-family:Symbol;">f</span> consists of the productions.<br /> S <span style="font-family:Symbol;">®</span> aSBC, S <span style="font-family:Symbol;">®</span> aBC, CB <span style="font-family:Symbol;">®</span> BC, aB <span style="font-family:Symbol;">®</span> ab, bB <span style="font-family:Symbol;">®</span> bb, bC <span style="font-family:Symbol;">®</span> bc, cC <span style="font-family:Symbol;">®</span> cc.<br /> <b> <br />Example 3:</b><br /> <br />G = ({S, A, B, C} , {a, b}, S , <span style="font-family:Symbol;">f</span> ), where <span style="font-family:Symbol;">f</span> consist of the set of productions,<br /> <br />S <span style="font-family:Symbol;">®</span> aS, S <span style="font-family:Symbol;">®</span> aB, B <span style="font-family:Symbol;">®</span> bC, C <span style="font-family:Symbol;">®</span> aC, C <span style="font-family:Symbol;">®</span> a. <br /> <br />Let G = (V<sub>N</sub>, V<sub>T</sub>, S, <span style="font-family:Symbol;">f</span> ) be a grammar. The string <span style="font-family:Symbol;">y</span> produces <span style="font-family:Symbol;">s</span> (<span style="font-family:Symbol;">s</span> reduces to <span style="font-family:Symbol;">y</span> or <span style="font-family:Symbol;">s</span> is the derivation of <span style="font-family:Symbol;">y</span> ), written as<br /><span style="font-family:Symbol;">y</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/darrplus.gif" width="34" align="absmiddle" border="0" height="20" /><span style="font-family:Symbol;">s</span> if there are strings <span style="font-family:Symbol;">f</span> <sub>0</sub>, <span style="font-family:Symbol;">f</span> <sub>1</sub>, … ,<span style="font-family:Symbol;">f</span> <sub>n</sub> ( n > 0) such that <span style="font-family:Symbol;">y</span> = <span style="font-family:Symbol;">f</span> <sub>0</sub> <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">f</span> <sub>1 </sub>, <span style="font-family:Symbol;">f</span> <sub>1</sub> <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">f</span> <sub>2</sub>, <sub> </sub>… , <span style="font-family:Symbol;">f</span> <sub>n-1</sub> <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">f</span> <sub>n</sub> and <span style="font-family:Symbol;">f</span> <sub>n</sub> <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">s</span> . <br /> <br />For example the string a<sup>2</sup> b<sup>2</sup>c<sup>2</sup> can be derived from the grammar given in Example 2.<br /> <br />For, S <span style="font-family:Symbol;">Þ</span> aSBC <span style="font-family:Symbol;">Þ</span> aaBCBC <span style="font-family:Symbol;">Þ</span> aabCBC <span style="font-family:Symbol;">Þ</span> aabBCC <span style="font-family:Symbol;">Þ</span> aabbCC <span style="font-family:Symbol;">Þ</span> aabbcC <span style="font-family:Symbol;">Þ</span> aabbcc = a<sup>2</sup>b<sup>2</sup>c<sup>2</sup>. <br /> <br /><b>Remark:</b> <br /> <br />Observe that as long as we have non-terminal character in the string, we can produce a new string from it. On the other hand, if a string contains only terminal symbols, then the derivation is complete and we cannot produce any further string from it. <br /> <br />A <i>sentential form</i> is any derivative of the unique non-terminal symbol S. The <i>language L</i> <i>generated by a grammar G</i> is the set of all sentential forms whose symbols are terminal. More precisely,<br /> <br /> L(G) = {<span style="font-family:Symbol;">s</span> / S <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/darrstar.gif" width="34" align="center" height="20" /> <span style="font-family:Symbol;">s</span> and <span style="font-family:Symbol;">s</span> <span style="font-family:Symbol;">Î</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sec251.gif" width="36" align="absmiddle" border="0" height="42" />}.<br /> <br /> That is, the language is a subset of the set of all terminal strings over V<sub>T</sub>.<br /> <br /> For example in the grammar G<sub>4</sub>= ({S,C}, {a, b}, S, <span style="font-family:Symbol;">f</span> ), where <span style="font-family:Symbol;">f</span> is the set of productions<br /> <br /> S <span style="font-family:Symbol;">®</span> aCa , C <span style="font-family:Symbol;">®</span> aCa , C <span style="font-family:Symbol;">®</span> b, has the sentential forms of form aabaa, aaabaaa, etc. In general L(G<sub>4</sub>) consists of the sentential forms of the form <i>a<sup>n</sup>ba<sup>n</sup>, </i> n <span style="font-family:Symbol;">³ </span>1<i>.</i> Since every sentential form has to start with the production rule S <span style="font-family:Symbol;">®</span> aCa, and the non-terminal symbol C can always be replaced either by the production C <span style="font-family:Symbol;">®</span> aCa or by C <span style="font-family:Symbol;">®</span> b. As long as we use the rule<br />C <span style="font-family:Symbol;">®</span> aCa the length of the sentential form will grow and it will have only one non-terminal symbol C and other terminal symbols are the symbol "a’s" appearing equally on either side of C. If we use the production C <span style="font-family:Symbol;">®</span> b, then the sentential form will consist of only terminal symbols, hence the derivation is complete. Thus all the sentential form are of the form a<sup>n</sup>ba<sup>n</sup>.<br /> <b> <br /> Remark:<br /> <br /> </b> It is clear that the language L generated by any grammar G = (V<sub>N</sub>, V<sub>T</sub>, S, <span style="font-family:Symbol;">f</span> ), will be a subset of the free semigroup on the alphabet V<sub>T</sub>.<br /> <br />There are four different types of grammars depending on the nature of its production rules.<br /> <br /> <i>A grammar is called <b>context-sensitive</b>, if all of its productions are of the form <span style="font-family:Symbol;">a</span> </i><span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">b</span> <i>with </i>| <span style="font-family:Symbol;">a</span> | <span style="font-family:Symbol;">£</span> | <span style="font-family:Symbol;">b</span> |.<br /> <br /> In the context sensitive grammar, <span style="font-family:Symbol;">a</span> and <span style="font-family:Symbol;">b</span> in the production <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">b</span> can be expressed as <span style="font-family:Symbol;">a</span> = <span style="font-family:Symbol;">f</span> <sub>1</sub>A<span style="font-family:Symbol;">f</span> <sub>2</sub> and <span style="font-family:Symbol;">b</span> = <span style="font-family:Symbol;">f</span><sub>1 </sub> <span style="font-family:Symbol;">y</span> <span style="font-family:Symbol;">f</span> <sub>2</sub><br />(<span style="font-family:Symbol;">f</span> <sub>1</sub> and / or <span style="font-family:Symbol;">f</span> <sub>2</sub> are possibly empty) where <span style="font-family:Symbol;">y</span> must be non-empty.<br /> <br />Thus, in the production <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">b</span> , A is replaced by <span style="font-family:Symbol;">y</span> in the context of <span style="font-family:Symbol;">f</span> <sub>1</sub> and <span style="font-family:Symbol;">f</span> <sub>2</sub> . That is the reason we call this grammar as context sensitive. <br /> <br /><i>Language generated by context-sensitive grammar is called <b>context sensitive language</b>. <br /> </i><b> <br /> Example:<br /> <br /> </b> The grammar given in Example 2 is context sensitive.<br /> <br /> Next important and most useful grammar in many high-level languages is context free grammar<b>. </b><i>A grammar is called <b>context-free grammar</b>, if all its production are of the form <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">b</span> with |<span style="font-family:Symbol;">a</span> | <span style="font-family:Symbol;">£</span> | <span style="font-family:Symbol;">b</span> | and <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">Î</span> V<sub>N</sub>.<br /> <br /> </i> In the context free grammar, the rewriting variable in a sentential form is rewritten regardless of other symbols in its context. That is the reason we call this grammar as context-free grammar.<br /> <i> <br /> Language generated by a context free grammar is called a <b>context-free language</b></i>.<br /> <br /> The grammar given in the Example 1 is context-free grammar.<br /> <br /> The other simple yet powerful type of grammar is regular grammar.<br /> <i><br /> A grammar is said to be <b>regular grammar</b> if all of its production rules are of the form<br /> <span style="font-family:Symbol;"><br /> a</span> <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">b</span> with | <span style="font-family:Symbol;">a</span> | <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.5/Image/Image19.gif" width="13" align="absmiddle" height="16" /> | <span style="font-family:Symbol;">b</span> |, <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">Î</span> V<sub>N</sub> and <span style="font-family:Symbol;">b</span> has the form aB or a, where a<span style="font-family:Symbol;">Î</span> V<sub>T</sub> and B<span style="font-family:Symbol;">Î</span> V<sub>N</sub> .<br /> <br /> </i> The grammar given in Example 3 is a regular grammar.<br /> <i> <br /> A language generated by a regular grammar is called <b>regular language</b>.<br /> <br /> A grammar is sometimes called <b>unrestricted grammar</b> and language generated by a unrestricted grammar is called <b>unrestricted language</b>.</i> Observe that every grammar is an unrestricted grammar.<br /> <br /> Observe from the Examples 1, 2 and 3 that every regular grammar is a context-free grammar, every context-free grammar is a context-sensitive grammar but the converse is not necessarily true.<br /> <br /> Let <i>T<sub>0 </sub>, T<sub>1</sub>, T<sub>2</sub></i> and <i>T<sub>3</sub></i> denote unrestricted, context-sensitive, context-free and regular grammar respectively. Let <i>L(T<sub>i</sub>) </i>denote the class of all languages generated by the grammar <i>T<sub>i</sub></i> , 0 <span style="font-family:Symbol;">£</span> i <span style="font-family:Symbol;">£</span> 3. Thus it follows from the above discussion that <b><i>L(T<sub>3</sub>) <span style="font-family:Symbol;">Ì</span> L(T<sub>2</sub>) <span style="font-family:Symbol;">Ì</span> L(T<sub>1</sub>) <span style="font-family:Symbol;">Ì</span> L(T<sub>0</sub>).<br /> </i> </b>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-85652191057977504472008-08-12T04:23:00.004-07:002008-12-23T01:55:02.555-08:00Direct Product of Semigroups<b><span style="font-size:130%;"><a name="2.4 Direct Product of the Semigroups"></a></span></b><br /> <br />From the given two algebraic structures we can always get a bigger algebraic structure by taking the cross product of the two<br /> structures.<br /> <br /> Let (S , *) and (T , <span style="font-family:Symbol;">D</span> ) be two semigroup. <i>The <b>direct product</b> of (S , *) and (T , <span style="font-family:Symbol;">D</span> ) is the algebraic structure (S<span style="font-family:Symbol;">´</span> T, </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i>),<br /> where the operation </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i> on S<span style="font-family:Symbol;">´</span> T is defined by (s<sub>1 </sub>, t<sub>1</sub>) </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i> (s<sub>2 </sub>, t<sub>2</sub>) = ( s<sub>1</sub> * s<sub>2 </sub>, t<sub>1</sub> <span style="font-family:Symbol;">D</span> t<sub>2</sub>), for any two pairs (s<sub>1 </sub>, t<sub>1</sub>) and<br />(s<sub>2</sub>, t<sub>2</sub>) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.4/Image/Image1.gif" width="12" height="12" /> S </i><span style="font-family:Symbol;">´</span><i>T.<br /> <br /><br /> </i>From the definition it follows that (S<span style="font-family:Symbol;">´</span> T, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup because the binary operation <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> in S<span style="font-family:Symbol;">´</span> T is defined in terms of the<br /> operations * and <span style="font-family:Symbol;">D</span> and both are associative, so the new operation ‘<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />’ is also associative in S<span style="font-family:Symbol;">´</span> T. Thus, (S<span style="font-family:Symbol;">´</span> T, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a<br /> semigroup. Further, if S and T both are monoids with e and e<span style="font-family:Symbol;">¢</span> be their respective identity elements then the element (e, e<span style="font-family:Symbol;">¢</span> ) of<br /> S<span style="font-family:Symbol;">´</span> T acts as an identity element. Hence (S<span style="font-family:Symbol;">´</span> T , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a monoid.Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-52996298935800128002008-08-12T04:23:00.003-07:002008-12-23T01:55:02.555-08:00Subsemigroups and Submonoids<b><nobr><span style="font-size:130%;"><a name="2.3 Subsemigroups and Submonoids"></a><br /><br /></span></nobr> </b><p align="left"><nobr>Studying the substructure always helps to understand the whole algebraic structure in depth and detail. Here in this section we<br /> discuss about some basic facts on the subsemigroups and submonoids.<br /> <br /> <i> Let (S , *) be a semigroup and T <span style="font-family:Symbol;">Í</span> S. If the set T is closed under the operation *, then</i> <i>(T , *) is said to be a <b><br /> subsemigroup</b> of (S , *).<br /><br />Similarly, let (M , * , e) be a monoid and T <span style="font-family:Symbol;">Í</span> M. If T is closed under the operation * and</i> <i>e <span style="font-family:Symbol;">Î</span> T, then (T , * , e) is said<br /> to be a <b>submonoid</b> of (M , * , e). </i><br /><br /> For example, (N , ·) is a subsemigroup of (Z , ·), the semigroup (E , +) is a subsemigroup of the semigroup of (N, +), where<br /> E is the set of all even positive integers. The semigroup (N , +) is not a submonoid of the monoid (Z , +).<br /> <b><br />Theorem 2.3.1:<br /> </b><br />Any nonempty intersection of subsemigroups S<sub>i</sub>, of a semigroup (S , *) is again a subsemigroup. In general union of<br /> subsemigroup of (S , *) is not necessarily a subsemigroup.<br /> <b><br />Proof:</b><br /><br />If x and y are in <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.3/Image/Image200.gif" width="29" align="absmiddle" height="36" /> S<sub>i</sub>, then x <span style="font-family:Symbol;">Î</span> S<sub>i</sub> and y <span style="font-family:Symbol;">Î</span> S<sub>i</sub>, for all i. Thus, x * y <span style="font-family:Symbol;">Î</span> S<sub>i</sub> , for all i.<br /> Hence, x * y <span style="font-family:Symbol;">Î</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.3/Image/Image200.gif" width="29" align="absmiddle" height="36" />S<sub>i</sub>. Therefore, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.3/Image/Image201.gif" width="29" align="absmiddle" height="36" />S<sub>i</sub> is a subsemigroup of (S , *).<br /><br />Consider the semigroup (Z<sub>6</sub> , +) and its subsemigroups, H<sub>1</sub> = ({2], [4], [0]}, +) and H<sub>2</sub> = ({[0], [3]} , +).<br /> Then (H<sub>1</sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.3/Image/Image3.gif" width="17" height="13" /> H<sub>2</sub> ,+ ) is not a subsemigroup, since [2] + [3] = [5], the equivalence class [5], but [5] is not a member of H<sub>1</sub><span style="font-family:Symbol;">È</span>H<sub>2</sub>.<br /><br />But if H<sub>1 </sub> <span style="font-family:Symbol;">Í</span> <sub> </sub>H<sub>2</sub> then H<sub>1</sub> <span style="font-family:Symbol;">È</span> H<sub>2</sub> = H<sub>2 </sub>, so, H<sub>1</sub> <span style="font-family:Symbol;">È</span> H<sub>2</sub> is a subsemigroup of (S , *).<br /> <b><br />Theorem 2.3.2:<br /> </b><br /> For any commutative monoid (M , * , e), the set of idempotent elements of M form a submonoid.<br /> <b><br />Proof:</b><br /><br />Let S be the set of all idempotent elements of (M , * , e). It is clear that e is an idempotent element, since e * e = e.<br /> Thus, S <span style="font-family:Symbol;">¹</span> <span style="font-family:Symbol;">f</span> .<br /><br />Let a, b <span style="font-family:Symbol;">Î</span> S. Then a * a = a and b * b = b.<br /> Now (a * b) * (a * b) = (a * b) * (b * a) (since M is commutative)<br /> = a * (b * b) * a (by associativity)<br /> = a * (b * a) (as b * b = b)<br /> = a * (a * b) (by commutative)<br /> = (a *a) * b (by associativity)<br /> = a * b (as a * a = a).<br /> Therefore S is closed under the operation *.<br /> Hence (S , *) is a submonoid of (M , * , e).<br /><br /><i>Let (S , *) be a semigroup and but <span style="font-family:Symbol;">f</span> <span style="font-family:Symbol;">¹</span> T <span style="font-family:Symbol;">Í</span> S. Then the intersection of all subsemigroup containing T forms a<br /> subsemigroup. This is the smallest subsemigroup, which contains T and is denoted by <t>. Also, we call <b><t><br /> the subsemigroup generated by T</b>. </i></nobr></p>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-16189506385224334092008-08-12T04:23:00.001-07:002008-12-23T01:55:02.555-08:00Homomorphism and Quotient Semigroup<span style="font-size:130%;"><b><a name="2.2 Homomorphism and Quotient Semigroup"> </a></b></span><br /> <br />The concept of homomorphism helps to understand the structural similarity between two given algebraic structures.<br /> <br /><i>Let (S , *) and (T , <span style="font-family:Symbol;">D</span> ) be any two semigroups.</i> <i>A mapping g : S <span style="font-family:Symbol;">®</span> T such that for any two elements a, b <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image1.gif" width="12" height="12" /> S,<br /> g (a * b) = g (a) <span style="font-family:Symbol;">D</span> g (b) is called <b>semigroup homomorphism</b>.</i><br /> <b> <br />Remark:</b><br /> <br />Homomorphism essentially preserves the behavior of the operations on the mapped elements.<br /> <br /> <i> <br />A semigroup homomorphism is called a semigroup <b>monomorphism, epimorphism or isomorphism</b> depending on<br /> whether the mapping is "one - one", "onto", or "one - one and onto " respectively</i>. <i>A semigroup homomorphism<br /> onto itself is called <b>endomorphism</b></i>.<br /> <i> <br />Two semigroups (S , *) and (T , <span style="font-family:Symbol;">D</span> ) are said to be<b> isomorphic</b> if there exists a semigroup isomorphism mapping from<br /> S onto T.<br /> </i><b> <br />Example 1:<br /> </b> <br />Consider the semigroup (Z , ·) and (N, ·). Define f : Z <span style="font-family:Symbol;">®</span> N by f(x) = |x|.<br /> Then, for all, x, y <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image1.gif" width="12" height="12" /> Z , f(x , y) = |x · y| = |x| · |y| = f(x). f(y).Thus f is homomorphism.<br /> f is also onto, since for every n <span style="font-family:Symbol;">Î</span> N, there exists –n, n <span style="font-family:Symbol;">Î</span> Z such that f(n) = |n| = n and f(-n) = |-n| = n.<br /> But, f is not one-one for, f(3) = f(-3) = 3, but 3 <span style="font-family:Symbol;">¹ </span>-3.<br /> <br /><b>Example 2:<br /> </b> <br />Define g : (Z , ·) <span style="font-family:Symbol;">®</span>(N<sub>0</sub> , ·) by g(x) = 0. Then g is homomorphism.<br /> <br />For, g(x · y) = g(xy) = 0 = 0 · 0 = g(x) · g(y), for all x , y <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image1.gif" width="12" height="12" /> Z.<br /> <b> <br />Example 3:<br /> </b> <br />Let (N<sub>0</sub> , +) be the semigroup of natural numbers and (S , *) be the semigroup on S ={e, 0, 1} with the operation *<br /> given by the following table.<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image186.gif" width="81" height="93" /><br /> <br />A mapping g : N<sub>0</sub> <span style="font-family:Symbol;">®</span> S given by g(0) = 1 and g( j ) = 0 for j <span style="font-family:Symbol;">¹</span> 0, is homomorphism,<br /> <br />for, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image187.gif" width="226" align="absmiddle" border="0" height="48" />.<br /> <br />Therefore from the table of (S , * ) , we have g(a + b) = g(a) * g(b).<br /> <br />Thus, g is a semigroup homomorphism. But g does not map the identity element 0 (zero) of the semigroup (N , +) onto the<br /> identity element e of (S , *).<br /> <br /> <br />From the Example 3 of semigroup homomorphism, one can observe that semigroup homomorphism not necessarily always<br /> maps an identity element onto identity element. Thus, we have the following definition for the monoid homomorphism. <br /> <br /> <i> Let, (M , * , e<sub>M</sub>) and (T , <span style="font-family:Symbol;">D</span> , e<sub>T</sub>) be any two monoids.</i> <i>A mapping </i>g : M <span style="font-family:Symbol;">®</span> T<i> such that for any two elements, </i>a, b <span style="font-family:Symbol;">Î</span> M<i> ,<br /> </i>g(a * b) = g(a) <span style="font-family:Symbol;">D</span> g(b) and g(e<sub>M </sub>) = e<sub>T<i> </i></sub><i>is called a <b>monoid homomorphism</b>.</i> <br /> <br />Example 1 of semigroup homomorphism is also monoid homomorphism, since it maps the identity element 1 of (Z , ·) onto<br /> the identify element 1 of (N , ·).<br /> <b> <br />Theorem 2.2.1 :<br /> </b> <br />Let (S , *) be a semigroup. Then H, the set of all endomorphism on S is a semigroup with respect to the operation composition<br /> of mapping.<br /> <b> <br />Proof :</b><br /> <br />Let H be the set of all endomorphism on a semigroup (S , *).<br /> Let f, g <span style="font-family:Symbol;">Î</span> H and let x, y <span style="font-family:Symbol;">Î</span> S.<br /> Then, (f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g) (x * y) = f (g(x * y))<br /> <br /> = f(g(x) * g(y))<br /> <br /> = f(g(x)) * f(g(y))<br /> <br /> = (f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g)(x) * (f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g)(y).<br /> Thus, f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g <span style="font-family:Symbol;">Î</span> H.<br /> Hence (H , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup, since the composition of mapping '<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />' is associative.<br /> <br /><b>Theorem 2.2.2 :<br /> </b><br />Let (S , *) be a given semigroup. There exits a homomorphism g : S <span style="font-family:Symbol;">®</span> S<sup>S</sup>, where (S<sup>S</sup>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup of mappings from S<br /> into S under the operation of composition of mappings.<br /> <b> <br />Proof :<br /> </b> <br />Let (S , *) be a given semigroup. For each a <span style="font-family:Symbol;">Î</span> S, define g : S <span style="font-family:Symbol;">®</span> S<sup>S</sup> by g(a) = f<sub>a </sub>, where f<sub>a</sub>(x) = a * x, for all x <span style="font-family:Symbol;">Î</span> S.<br /> Then, f<sub>a * b </sub> (x) = (a * b) * x<br /> = a * ( b * x)<br /> = f<sub><span style="font-family:b,Times New Roman;">a</span></sub> ( b * x)<br /> = f<sub><span style="font-family:b,Times New Roman;">a</span></sub>(f<span style="font-family:b,Times New Roman;"><sub>b</sub></span>(x))<br /> = f<sub><span style="font-family:b,Times New Roman;">a</span></sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> f<sub>b</sub> (x), for all x <span style="font-family:Symbol;">Î</span> S.<br /> Thus, f<sub>a </sub><sub>* b</sub> = f<sub><span style="font-family:b,Times New Roman;">a</span></sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> f<sub>b</sub> .<br /> Now, g(a * b) = f<sub>a*b</sub> = f<sub><span style="font-family:b,Times New Roman;">a</span></sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> f<sub>b</sub> = g(a) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g(b).<br /> Hence, g is a semigroup homomorphism from S into S<sup>S</sup>.<br /> <b> <br />Remark 1:<br /> </b> <br />Let g(S) denote the image set of S under the mapping g. Therefore by the Theorem 2.2.2, g (S) <span style="font-family:Symbol;">Í</span> S<sup>S</sup>. That is, S can be<br /> identified as a substructure of S<sup>S</sup> . Therefore every semigroup can be visualized as a substructure of S<sup>S</sup>.<br /> <b> <br />Theorem 2.2.3:<br /> </b> <br />Let (M , * , e) be a monoid. Then there exists a subset T <span style="font-family:Symbol;">Í</span> M <sup>M</sup> such that (M , * , e) is isomorphic to the monoid T.<br /> <b> <br />Proof:</b><br /> <br />For any element a <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image1.gif" width="12" height="12" /> M , define g (a) = f <sub><span style="font-family:b,Times New Roman;">a</span></sub> , where f <sub><span style="font-family:b,Times New Roman;">a</span></sub> (x) = a * x, for all x <span style="font-family:Symbol;">Î</span> M.<br /> Then f<sub><span style="font-family:b,Times New Roman;">a</span></sub> <span style="font-family:Symbol;">Î</span> M<sup>M</sup> .<br /> Let T = {f<span style="font-family:b,Times New Roman;"><sub>a </sub> <span style="font-family:Symbol;">Î</span> <sub> </sub> </span><span style="font-family:b,Times New Roman;">M<sup>M</sup><sub> </sub>/</span> f<sub>a</sub>=a * x , for all a <span style="font-family:Symbol;">Î</span> M and for all x <span style="font-family:Symbol;">Î</span> M }.<br /> Then by the Theorem 2.2.1, g is a homomorphism from M into T.<br /> Also, for each f<sub><span style="font-family:b,Times New Roman;">a</span></sub> <span style="font-family:Symbol;">Î</span> T, there exist a <span style="font-family:Symbol;">Î</span> M such that g(a) = f<sub><span style="font-family:b,Times New Roman;">a</span></sub><span style="font-family:b,Times New Roman;">.<br /> </span>Thus, g is onto.<br /> <br />Further, if a <span style="font-family:Symbol;">¹</span> b, then g(a) <span style="font-family:Symbol;">¹</span> g(b). For, suppose g(a) = g(b), then a * x = b * x, for all x <span style="font-family:Symbol;">Î</span> M, in particular for x <span style="font-family:Symbol;">Î</span> e also.<br /> Therefore, a * e = b * e, that is, a = b, a contradiction.<br /> Thus, g (a) <span style="font-family:Symbol;">¹</span> g (b).<br /> Hence, g is one - one.<br /> So, g is isomorphism from M to T.<br /> Hence, M <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image188.gif" width="15" height="12" /> T.<br /> <b> <br />Theorem 2.2.4:<br /> </b> <br />Let X be a set containing n elements, let X* denote free semigroup generated by X, and let (S , <span style="font-family:Symbol;">Å</span> ) be any other semigroup<br /> generated by any n generators. Then there exists a homomorphism g : X<sup>*</sup> <span style="font-family:Symbol;">®</span> S.<br /> <b> <br />Proof:</b><br /> <br />Let Y be the set of n generators of S, consisting of y<sub>1</sub>,y<sub>2</sub>, … ,y<sub>n</sub> and let x<sub>1</sub>, x<sub>2</sub>, … ,x<sub>n</sub> be the n elements of X.<br /><br />Define g(x<sub>i</sub>) = y<sub>i, </sub>for i = 1,2, … ,n.<br />Then, for any string <span style="font-family:Symbol;">a</span> = x<sub>1</sub>, x<sub>2</sub> … x<sub>n</sub> of X*, we define g(<span style="font-family:Symbol;">a</span> ) = g(x<sub>1</sub>) <span style="font-family:Symbol;">Å</span> g(x<sub>2</sub>) <span style="font-family:Symbol;">Å</span> … <span style="font-family:Symbol;">Å</span> g(x<sub>m</sub>) .<br />Then, for any <span style="font-family:Symbol;">b</span> = <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image189.gif" width="18" align="absmiddle" height="22" /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image190.gif" width="20" align="absmiddle" height="22" /> … <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image191.gif" width="19" align="absmiddle" height="24" />, we have<br />g (<span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">b</span> ) = g (x<sub>1</sub>x<sub>2</sub> … x<sub><span style="font-family:b,Times New Roman;">m</span></sub><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image189.gif" width="18" align="absmiddle" height="22" /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image190.gif" width="20" align="absmiddle" height="22" /> … <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image191.gif" width="19" align="absmiddle" height="24" />)<br /> = g(x<sub>1</sub>) <span style="font-family:Symbol;">Å</span> g(x<sub>2</sub>) <span style="font-family:Symbol;">Å</span> … <span style="font-family:Symbol;">Å</span> g(x<sub>m</sub>) <span style="font-family:Symbol;">Å</span> g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image189.gif" width="18" align="absmiddle" height="22" />) <span style="font-family:Symbol;">Å</span> g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image190.gif" width="20" align="absmiddle" height="22" />) <span style="font-family:Symbol;">Å</span> … <span style="font-family:Symbol;">Å</span> g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image191.gif" width="19" align="absmiddle" height="24" />)<br /> = g(<span style="font-family:Symbol;">a</span> ) <span style="font-family:Symbol;">Å</span> g(<span style="font-family:Symbol;">b</span> ) .<br />Thus, g is a homomorphism.<br />Hence the theorem.<br /><br /> <br /> <i> Let (X , </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i>) be an algebraic system in which </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i> is a binary operation on X. An equivalence relation E on X is said<br /> to have the <b>substitution property</b> with respect to the operation </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i> for any x<sub>1 </sub>, x<sub>2 </sub><span style="font-family:Symbol;">Î</span><sub> </sub>X.<br /> (x<sub>1</sub> E <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" />) and (x<sub>2</sub></i> <i>E <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" />) then (x<sub>1</sub> </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i> x<sub>2</sub> ) E (<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image194.gif" width="18" align="absmiddle" height="22" /> </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image195.gif" width="20" align="absmiddle" height="22" /> ), where <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image196.gif" width="18" align="absmiddle" height="22" /> , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image197.gif" width="20" align="absmiddle" height="22" /> <span style="font-family:Symbol;">Î</span> X.<br /> </i>That is, if x<sub>1</sub> is related to <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image196.gif" width="18" align="absmiddle" height="22" /> and x<sub>2</sub> is related to <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image197.gif" width="20" align="absmiddle" height="22" />, then the result of x<sub>1</sub> and x<sub>2</sub> by the binary operation <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />, i.e., x<sub>1</sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> x<sub>2 </sub>,<br /> is related to <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image196.gif" width="18" align="absmiddle" height="22" /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image197.gif" width="20" align="absmiddle" height="22" />, the result of the binary operation <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> on the related elements <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> and<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image197.gif" width="20" align="absmiddle" height="22" />. More precisely, x<sub>1</sub><sup> </sup> is related<br /> to <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> and x<sub>2 </sub> is related to<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" /><sup> </sup>then x<sub>1</sub> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> x<sub>2 </sub>can be equivalently substituted for <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" /> .<br /> <i> <br />Let (X , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /></i><i>) be an algebraic system and E be an equivalence relation on X. The relation E is called a <b>congruence<br /> relation</b> on (X , </i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><i>) if E satisfies the substitution property with respect to the operation ‘</i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />’<i>.<br /> </i><b> <br />Theorem 2.2.5:<br /> </b> <br />Let (S , *) and (T , <span style="font-family:Symbol;">D</span> ) be two semigroups and g be a semigroup homomorphism from (S , *) to (T , <span style="font-family:Symbol;">D</span> ). Corresponding to<br /> the homomorphism g, there exists a congruence relation R on (S , *) defined by xRy, if g(x) = g(y) for x, y <span style="font-family:Symbol;">Î</span> S.<br /> <b> <br />Proof:<br /> <i> <br />The relation R defined above is an equivalence relation:<br /> <br /></i></b>Since g(x) = g(x), for all x <span style="font-family:Symbol;">Î</span> S, xRx, for all x <span style="font-family:Symbol;">Î</span> S.<br /> Therefore, R is reflexive.<br /> Also, if g(x) = g(y), then g(y) = g(x).<br /> That is, if xRy then yRx .<br /> Therefore, R is symmetry.<br /> Further, if g(x) = g(y) and g(y) = g(z), then g(x) = g(y) = g(z), that is, g(x) = g(z) .<br /> That is, if xRy and yRz then xRz.<br /> Therefore, R is transitive.<br /> Thus, R is an equivalence relation . <br />Let x<sub>1</sub>, x<sub>2 </sub>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" />and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" /> <span style="font-family:Symbol;">Î</span> S such that xR<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> and x<sub>2</sub>R<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" />.<br /> Then g(x<sub>1</sub> * x<sub>2</sub>) = g(x<sub>1</sub>) <span style="font-family:Symbol;">D</span> g(x<sub>2</sub>) (since g is a homomorphism.)<br /> = g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" />) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image198.gif" width="15" height="17" />g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" />) (since x<sub>1</sub>R<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> and x<sub>2</sub>R<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" />, so g(x<sub>1</sub>) = g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" />) and g(x<sub>2</sub>) = g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" />)]<br /> = g(<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> * <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" />) (since g is a homomorphism.).<br /> Therefore, by definition of R, x<sub>1</sub> * x<sub>2</sub> R<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image192.gif" width="17" align="absmiddle" height="22" /> * <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.2/Image/Image193.gif" width="20" align="absmiddle" height="22" /> <i><span style="font-family:b,Times New Roman;">.</span></i><br /> Thus, R is congruence relation on (S , *)<span style="font-size:85%;"> . <br /> </span><b> <br />Theorem 2.2.6:<br /> </b> <br />Let (S , *) be a semigroup and R be a congruence relation on (S , *) on the quotient set<br />S/R = {[a] / a <span style="font-family:Symbol;">Î</span> S}, define [a] <span style="font-family:Symbol;">Å</span> [b] = [a * b], for all [a], [b] <span style="font-family:Symbol;">Î</span> S/R . Then (S/R , <span style="font-family:Symbol;">Å</span> ) is a semigroup. Further, there exists a<br /> homomorphism from (S , *) onto (S/R , <span style="font-family:Symbol;">Å</span> ) called natural homomorphism.<br /> <b> <br />Proof:<br /> </b> <br />Let (S, *) be given semigroup and let [a] denote an equivalence class of a corresponding to the congruence relation R. By the<br /> Theorem 2.2.5 such congruence relation exists.<br /> <b> <br />Claim 1 : </b><i>The operation <span style="font-family:Symbol;">Å</span> is well – defined.</i><br /> <br /> For, if [a] = [a'] and [a] = [b'], then aRa' and bRb' .<br /> Since R is congruence relation, we have, a * b R a' * b' .<br /> Thus, [a * b] = [a' * b'] .<br /> <br /> Consequently, [a] <span style="font-family:Symbol;">Å</span> [b] = [a * b] = [a' * b'] = [a'] <span style="font-family:Symbol;">Å</span> [b'].<br /> Hence the Claim 1.<br /> <br /> <b>Claim 2 :</b> <i>(S/R ,<span style="font-family:Symbol;">Å</span> ) is a semigroup.</i><br /> <br /> For all [a], [b], [c] <span style="font-family:Symbol;">Î</span> S/R, we have<br /> [a] <span style="font-family:Symbol;">Å</span> ([b] <span style="font-family:Symbol;">Å</span> [c] ) = [a] <span style="font-family:Symbol;">Å</span> [ b * c]<br /> = [a * ( b * c)]<br /> = [ (a * b) * c ]<br /> = [ a * b] <span style="font-family:Symbol;">Å</span> [c]<br /> = ( [a] <span style="font-family:Symbol;">Å</span> [b] ) <span style="font-family:Symbol;">Å</span> [c].<br /> That is, <span style="font-family:Symbol;">Å</span> is associative.<br /> Hence (S/R , <span style="font-family:Symbol;">Å</span> ) is a semigroup.<br /> Hence the Claim 2.<br /> <b> <br />Claim 3</b> : <i>There exists a natural homomorphism.</i><br /> <br /> Define a mapping g : S <span style="font-family:Symbol;">®</span> S/R by g(a) = [a] ,for every a <span style="font-family:Symbol;">Î</span> S .<br /> Then, g(a* b) = [a * b] = [a] <span style="font-family:Symbol;">Å</span> [b] = g(a) <span style="font-family:Symbol;">Å</span> g(b) ,for all a , b <span style="font-family:Symbol;">Î</span> S.<br /> Thus, g is a homomorphism.<br /> <br /> Further, for each [a] <span style="font-family:Symbol;">Î</span> S/R there exist a <span style="font-family:Symbol;">Î</span> S such that g(a) = [a].<br /> Thus, g is a homomorphism from (S , *) onto (S/R , <span style="font-family:Symbol;">Å</span> ) .<br /> Hence the Claim 3.<br /> <br />Hence the theorem.Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-44109030842404810542008-08-12T04:22:00.002-07:002008-12-23T01:55:02.555-08:00Semigroups and Monoids<b><nobr><span style="font-size:130%;"><a name="2.1 Semigroups and Monoids"> </a><br /> </span> </nobr> </b> <p align="left"><nobr> <br /><br /> <br />Semigroups play a fundamental role in the Algebraic Automata Theory and the Theory of Formal Languages. In this chapter<br /> we discuss introductory results on semigroups, monoids and grammars and some popular examples..<br /> <span style="font-size:85%;"><br /> </span> <i>A nonempty set S together with a binary operation *, (S, *), is called a <b>semigroup</b>, if for all a, b, c<span style="font-family:Symbol;">Î</span> S,<br /> a * (b*c) = (a * b) * c.<br /> </i> <br />Examples of Semigroups:<br /><ol><li> (N , +) , (N , ·), (Z , +), (Z , ·), (Q , +), (Q , ·), (R , +), (R , ·), (C , +) and (C , ·) are all semigroups, where N, Z, Q, R<br /> and C respectively denote the set of natural numbers, the set of integers, the set of rational numbers, the set of real<br />numbers and the set of complex numbers and + and · denote usual addition and usual multiplication of number. [It is<br />clear that + and · are associative on the above number system].</li><li> (R, max), where R the set of real number and x max y = max (x, y) is a semigroup.<br /> For, (x max y) max z = (max (x , y)) max z<br /> = max (max (x , y), z)<br /> = max (x , y , z)<br /> = max (x, max (y, z))<br /> = x max (max (y , z))<br /> = x max (y max z) , for all x, y, z <span style="font-family:Symbol;">Î</span> R.<br /> Thus, (R, max) is a semigroup.</li><li> Let S be any nonempty set. Let S<sup>S</sup> be the set of all mappings from S to S and let <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />denote the composition of mappings.<br /> Then (S<sup>S </sup>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup.</li></ol> <blockquote><b>Proof:</b><br /><br />Let S be any nonempty set.<br />Let S<sup>S</sup> = {f / f: S <span style="font-family:Symbol;">®</span> S be any mapping}.<br />Then, the composition of mapping f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g is defined by<br />(f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g)(x) = f(g(x)), for all x <span style="font-family:Symbol;">Î</span> S, is assosiative.<br />For, (f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> h)(x) = (f <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /> g) (h(x)) = f(g(h(x))), for all x <span style="font-family:Symbol;">Î</span> S and<br />on the other hand, (f<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />(g<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />h)) (x) = f((g<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />h)(x)) = f(g(h(x))), for all x <span style="font-family:Symbol;">Î</span> S.<br />Thus, (f<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />g) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />h = (f<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />g) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />h, for all f, g, h <span style="font-family:Symbol;">Î</span> S<sup>S</sup>.<br />Hence, (S<sup>S</sup>,<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup.<br /></blockquote> <ol start="4"><li> Let S be any set. Then (<i>P</i>(S) , <span style="font-family:Symbol;">Ç</span> ) is a semigroup, where <i>P</i>(S) denote the power set of S. Let A, B and C be any three<br /> subsets of S. Then, (A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">Ç</span> C = A <span style="font-family:Symbol;">Ç</span> (B <span style="font-family:Symbol;">Ç</span> C).<br /> For, x <span style="font-family:Symbol;">Î</span> (A <span style="font-family:Symbol;">Ç</span> B) <span style="font-family:Symbol;">Ç</span> C <span style="font-family:Symbol;">Û</span> x <span style="font-family:Symbol;">Î</span> A <span style="font-family:Symbol;">Ç</span> B and x <span style="font-family:Symbol;">Î</span> C <br /> <span style="font-family:Symbol;">Û</span> x <span style="font-family:Symbol;">Î</span> A and x <span style="font-family:Symbol;">Î</span> B and x <span style="font-family:Symbol;">Î</span> C<br /> <span style="font-family:Symbol;"> Û</span> x <span style="font-family:Symbol;">Î</span> A and x <span style="font-family:Symbol;">Î</span> B <span style="font-family:Symbol;">Ç</span> C<br /> <span style="font-family:Symbol;"> Û</span> x <span style="font-family:Symbol;">Î</span> A <span style="font-family:Symbol;">Ç</span> (B <span style="font-family:Symbol;">Ç</span> C).<br />Hence, (<i>P</i>(S) , <span style="font-family:Symbol;">Ç</span> ) is a semigroup.</li><li> (Z , -) is not a semigroup since the operation "-" is not associative.<br />For, (5 - 3) - 2 = 2 - 2 = 0, where as 5 - (3 - 2) = 5 - 1 = 4, definitely, (5 - 3) - 2 <span style="font-family:Symbol;">¹</span>5 - (3 - 2).</li></ol> Let (S , *) be a semigroup.<br /><ol type="i"><li> <i>An element s in (S , *) is called <b>zero element</b> of (S , *), if</i> <i>x * s = s * x = s holds for all x <span style="font-family:Symbol;">Î</span> S. </i></li><li> <i>An element e of (S , *) is called an <b>identity element</b> of (S , *) provided</i> <i>that s * e = e * s = s holds for all s <span style="font-family:Symbol;">Î</span> S.</i></li><li> <i>s is called an<b> idempotent</b> of (S , *), if s * s = s. </i></li><li> <i>An element s in (S ,* ) is said to be an <b>invertible</b> if there exists an </i> <i>element r in (S , *) such that s * r = e = r * s.</i></li><i> </i></ol> <i> </i> <i> A semigroup (S , *) is called a <b>monoid</b> if there is an identity element ‘e’ in (S , *).<br /> </i> <br />Generally we denote a monoid by a triple (M , * , e)<br /> <span style="font-size:85%;"><br /> </span><br /> <br /><b>Examples of Monoids<br /> </b> <ol><li> In the Example1 of the semigroups given above, except the semigroup (N , +), all the other semigroups are monoids. Since<br /> all the number system except N, have identity elements 0 and 1 respectively for the operations + and ·.</li><li> The semigroup (R , max) is not a monoid, since there is no identity element.</li><li> The semigroup (S<sup>S</sup> , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a monoid, since the identity mapping is the identity element of (S<sup>S</sup> , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />).</li><li> The semigroup (<i>P</i>(S) , <span style="font-family:Symbol;">Ç</span> ) for any set S is a monoid, where the set S acts as the identity element of (<i>P</i>(S) , <span style="font-family:Symbol;">Ç</span> ). </li><li> Let M<sub>n</sub> denote the set of all n x n matrices over the real numbers. Then (M<sub>n </sub>, +) and (M<sub>n</sub> , ·) are monoids, where ‘+’ is the<br /> matrix addition and ‘·’ is the matrix multiplication. Zero matrix acts as additive identity and unit matrix acts as multiplicative<br /> identity.</li></ol> The following two examples of monoid deserve a special attention.<br /> <b> <br />Example 6: Monoid of "Integer congruence Modulo"<br /> </b> <br />Let Z be the set of all integers. Define a relation "<span style="font-family:Symbol;">º</span> (mod m)" on the set of integers Z by a <span style="font-family:Symbol;">º</span> b (mod m) if and only if<br /> "m divides (a – b)", that is, "m<span style="font-family:Symbol;">½</span>(a – b)".<br /> Then, "<span style="font-family:Symbol;">º</span> (mod m)" is reflexive.<br /> For, any fixed m, m<span style="font-family:Symbol;">½</span>(a – a), for all a<span style="font-family:Symbol;">Î</span> Z.<br /> Therefore, a <span style="font-family:Symbol;">º</span> a (mod m).<br /> Also, "<span style="font-family:Symbol;">º</span> (mod m)" is symmetry.<br /> For, if a <span style="font-family:Symbol;">º</span> b (mod m), then m<span style="font-family:Symbol;">½</span>(a – b).<br /> Therefore, m<span style="font-family:Symbol;">½</span>-(a - b), that is, m<span style="font-family:Symbol;">½</span>(b – a).<br /> Thus, b <span style="font-family:Symbol;">º</span> a (mod m).<br /> Further, if a <span style="font-family:Symbol;">º</span> b (mod m) and b <span style="font-family:Symbol;">º</span> c (mod m), then m<span style="font-family:Symbol;">½</span>(a – b) and m<span style="font-family:Symbol;">½</span>(b – c) .<br /> Therefore, m<span style="font-family:Symbol;">½</span>((a - b) + ( b - c)) , that is, m<span style="font-family:Symbol;">½</span>(a – c).<br /> Thus, a <span style="font-family:Symbol;">º</span> c (mod m).<br /> Hence, "<span style="font-family:Symbol;">º</span> (mod m)" is transitive.<br /> Hence, "<span style="font-family:Symbol;">º</span> (mod m)" is an equivalence relation.<br /> <br />Let [a] = {x <span style="font-family:Symbol;">Î</span> Z / x <span style="font-family:Symbol;">º</span> a (mod m) }, denote the equivalence class of a <span style="font-family:Symbol;">Î</span> Z defined by the equivalence relation "<span style="font-family:Symbol;">º</span> (mod m)".<br /> <br />Let Z<sub>m</sub> denote the set of all distinct equivalence classes defined by the equivalence relation "<span style="font-family:Symbol;">º</span> (mod m)" [For, example Z<sub>3</sub> <br /> consists of the distinct equivalence classes [0], [1] and [2]]<br /> <br />Define the operations <span style="font-family:Symbol;">Å</span> and <span style="font-family:Wingdings;">¤</span> on the set Z<sub>m</sub> by [a] <span style="font-family:Symbol;">Å</span> [b] = [ (a + b) (mod m)] and [a] <span style="font-family:Wingdings;">¤</span> [b] = [(a · b) (mod m)].<br /> Then for all, [a], [b] and [c] in Z<sub>m</sub>, we have,<br /> ([a] <span style="font-family:Symbol;">Å</span> [b]) <span style="font-family:Symbol;">Å</span> [c] = [(a + b)(mod m)] <span style="font-family:Symbol;">Å</span> [c] (by definition of <span style="font-family:Symbol;">Å</span> )<br /> = [((a + b)(mod m) + c)(mod m)]<br /> = [(a + b + c)(mod m)] (by definition of 'mod m')<br /> = [(a + (b + c)(mod m)] (since + is associate in the integers)<br /> = [(a + ((b + c)(mod m))(mod m)]<br /> = [a] + [(b + c)(mod m)]<br /> = [a] + ([b] + [c])<br /> <br />Similarly, we can show that ([a] <span style="font-family:Wingdings;">¤</span> [b]) <span style="font-family:Wingdings;">¤</span> [c] = [a] <span style="font-family:Wingdings;">¤</span> ([b] <span style="font-family:Wingdings;">¤</span> [c]), for all, [a], [b], [c] <span style="font-family:Symbol;">Î</span> Z<sub>m</sub>.<br /> <br />Also, note that the equivalence class [0] act as an identity element in (Z<sub>m </sub>, <span style="font-family:Symbol;">Å</span> ) and the equivalence class [1] act as an identity<br /> element in (Z<sub>m </sub>, <span style="font-family:Wingdings;">¤</span> ).<br /> <br /><br /> <b>Example 7: Monoids of strings over an alphabet<br /> </b> <br /> Let <span style="font-family:Symbol;">å</span> be a nonempty set, may be finite or countable, called <b><i>alphabet</i></b>. The elements of <span style="font-family:Symbol;">å</span> are called <b><i>symbols</i></b><i> </i>or<i> <b>letters</b> </i>or<i> <b><br /> characters</b></i>. A <b><i>string</i></b><i> </i>or<i> <b>word</b></i> over an alphabet <span style="font-family:Symbol;">å</span> is an ordered set of symbols from the alphabet. A string consisting of m<br /> symbols (m > 0) is called a <b><i>string of length m</i></b>. A string of length zero, that is, string consisting of no symbols is called <b><i><br /> empty string</i></b> and it is denoted by <span style="font-family:Symbol;">L</span> . Let <span style="font-family:Symbol;">å</span> <b><i>*</i></b> denote the set of all strings over <span style="font-family:Symbol;">å</span> and let <span style="font-family:Symbol;">å</span> <b><i><sup>+</sup></i></b> denote the set of all nonempty<br /> strings over <span style="font-family:Symbol;">å</span> .<br /> <br />Let <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <span style="font-family:Symbol;">Î</span> <span style="font-family:Symbol;">å</span> * . Define a binary operation ‘<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />’called concatenation operation on strings by <span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">b</span> = <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">b</span><br /> <br />[For example, <span style="font-family:Symbol;">a</span> = aabbab and <span style="font-family:Symbol;">b</span> = abbbaa over <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit2/Section2.1/Image/Image184.gif" width="12" height="22" /><span style="font-family:Symbol;">å</span> = {a, b}, then <span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">b</span> = <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">b</span> = aabbababbbaa].<br /> <br />Then (<span style="font-family:Symbol;">å</span> * , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a monoid and (<span style="font-family:Symbol;">å</span> <sup>+</sup> , <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup.<br /> For <span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />(<span style="font-family:Symbol;">b</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">g</span> ) = <span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />(<span style="font-family:Symbol;">b</span> <span style="font-family:Symbol;">g</span> ) = <span style="font-family:Symbol;">a</span> (<span style="font-family:Symbol;">b</span> <span style="font-family:Symbol;">g</span> ) = (<span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">b</span> )<span style="font-family:Symbol;">g</span> = (<span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">b</span> )<span style="font-family:Symbol;">g</span> = (<span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">b</span> )<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">g</span> , for all <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g</span> <span style="font-family:Symbol;">Î</span> <span style="font-family:Symbol;">å</span> *.<br /> <br />Note that the empty string <span style="font-family:Symbol;">L</span> acts as an identity element since <span style="font-family:Symbol;">L</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">a</span> <sub> </sub>= <span style="font-family:Symbol;">a</span><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" /><span style="font-family:Symbol;">L</span> = <span style="font-family:Symbol;">a</span> , for all <span style="font-family:Symbol;">a</span> <span style="font-family:Symbol;">Î</span> <span style="font-family:Symbol;">å</span> *.<br /> Hence (<span style="font-family:Symbol;">å</span> <sup>+</sup>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a semigroup and (<span style="font-family:Symbol;">å</span> <sup>*</sup>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is a monoid.<br /> <i> <br />The semigroup (<span style="font-family:Symbol;">å</span> <sup>+</sup>,<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is also called <b>free semigroup</b> and (<span style="font-family:Symbol;">å</span> <sup>* </sup>, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Image/sdtembt.gif" width="15" border="0" height="20" />) is also called <b>free monoid.<br /> </b> <br /> </i>The free semigroup and free monoid play an important role in the "Theory of Formal Languages and Automata". <br /> <br /> <i> A semigroup (monoid) (S , *) is called <b>commutative</b> if a * b = b * a, for all a, b <span style="font-family:Symbol;">Î</span> S. <br /> <br />If in a monoid (M , * , e) every element is invertible, then the monoid is called <b>group</b>. <br /> <br /></i>For example, (Z , +) is commutative monoid and it is a group also. The monoid (M<sub>n </sub>, ·) is not commutative, and it is not a<br /> group. <br /> <br />In a monoid (M , *), the powers of any particular element say a <span style="font-family:Symbol;">Î</span> M are defined as<br /> a<sup>0</sup> = e, a<sup>1</sup> = a, a<sup>2</sup> = a * a, … , a<sup> j +1</sup> = a <sup>j</sup> * a, for j <span style="font-family:Symbol;">Î</span> N. <br /> <i> <br />A monoid (M , * , e) is said to be <b>cyclic</b> if there exists an element a <span style="font-family:Symbol;">Î</span> M such that every element of M can be expressed<br /> as some power of a, that is, as a<sup>n</sup> for some n <span style="font-family:Symbol;">Î</span> N</i>. <i>In such a case, the cyclic monoid is said to be generated by the<br /> element ‘a’ and the element ‘a’ is called <b>generator of the cyclic monoid</b>. <br /> </i><b> <br /><br /> Remark:</b><br /> <br />A cyclic monoid is always commutative, since, for any b, c <span style="font-family:Symbol;">Î</span> M, b = a<sup>m</sup> and c = a<sup>n</sup> for some m, n <span style="font-family:Symbol;">Î</span> N, thus, we have,<br /> b * c = a<sup>m</sup> * a<sup>n</sup> = a<sup>m+n</sup> = a<sup>n+m</sup> = a<sup>n</sup> * a<sup>m</sup> = c * b.<br /> <br /><b>Examples:</b> <br /> <ol><li> The semigroup (N , + ) is a cyclic semigroup, since (N , +) is generated by 1, that is, (N , +) = (1).</li><li> <br />The semigroup (N, ·) is a cyclic monoid, since any natural number can be expressed as product of powers of primes,<br /> that is (N , ·) is a cyclic monoid generated by the set of all primes and 1.</li></ol></nobr></p>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1tag:blogger.com,1999:blog-7399621945608861143.post-29799560889917378262008-08-12T04:22:00.001-07:002008-12-23T01:55:02.555-08:00Inference Theory of the Predicate Calculus<b><a name="Inference Theory of the Predicate Calculus"></a></b><br /><br />We use the concepts of equivalence and implication to formulas of the predicate calculus.<br /><br /><b><a name="1.6.1 Valid Formulas and Equivalences">1.6.1 Valid Formulas and Equivalences</a><br /></b><br />The formulas of the predicate calculus are assumed to contain statement variables, predicates, and object variables. The object variables are assumed to belong to a set called the universe of discourse or the domain of the object variable. Such a universe may be finite or infinite. The term 'variable' includes constants as a special case. In a predicate formula, when all the object variables are replaced by definite names of objects and the statement variables by statements, we obtain a statement which has a truth value T or F.<br /><br />The formulas of predicate calculus as given here do not contain predicate variables. They contain predicates; i.e., every predicate letter is intended to be a definite predicate, and hence is not available for substitution.<br /><br />Let A and B be any two predicate formulas defined over a common universe denoted by the symbol E. If, for every assignment of object names from the universe of discourse E to each of the variables appearing in A and B, the resulting statements have the same truth values, then the predicate formulas A and B are said to be equivalent to each other over E. This idea is symbolized by writing A<span style="font-family:Symbol;">Û</span>B over E. If E is arbitrary, then we say that A and B are equivalent, that is A<span style="font-family:Symbol;">Û</span> B.<br /><br />It is possible to determine by truth table methods whether a formula is valid in E, where E is a finite universe of discourse. This method may not be practical when the number of elements in E is large. It is impossible when the number of elements in E is infinite.<br /><br />Formulas of the predicate calculus that involve quantifiers and no free variables are also formulas of the statement calculus. Therefore, substitution instances of all the tautologies by these formulas yield any number of special tautologies.<br /><br />Consider the tautologies of the statement calculus given by P<span style="font-family:Symbol;">Ú</span><span style="font-family:Symbol;">ù</span>P, P<span style="font-family:Symbol;">®</span>Q<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /><span style="font-family:Symbol;">ù</span> P<span style="font-family:Symbol;">Ú</span>Q and substitute the formulas (x)R(x) and<br />(<span style="font-family:Symbol;">$</span> x)S(x) for P and Q respectively. It is assumed that (x)R(x) and (<span style="font-family:Symbol;">$</span>x)S(x) do not contain any free variables. The following tautologies are obtained.<br /><br />((x)R(x))<span style="font-family:Symbol;">Ú</span><span style="font-family:Symbol;">ù</span>((x)R(x)).<br /><br />((x)R(x))<span style="font-family:Symbol;">®</span>((<span style="font-family:Symbol;">$</span>x)S(x))<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /><span style="font-family:Symbol;">ù</span>((x)R(x))<span style="font-family:Symbol;">Ú</span>((<span style="font-family:Symbol;">$</span>x)S(x)).<br /><br />Let A(x), B(x), and C(x, y) denote any prime formulas of the predicate calculus. Then the following are valid formulas of the predicate calculus<br /><br /><span style="font-family:Symbol;"> ù ù</span>A(x)<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" />A(x).<br /><br /> C(x,y)<span style="font-family:Symbol;">Ù</span>B(x)<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" />B(x)<span style="font-family:Symbol;">Ù</span>C(x, y).<br /><br /> A(x)<span style="font-family:Symbol;">®</span>B(x)<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /><span style="font-family:Symbol;">ù</span>A(x)<span style="font-family:Symbol;">Ú</span>B(x).<br /><span style="font-size:85%;"> </span><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Section1.6.htm#1.6%20Inference%20Theory%20of%20the%20Predicate%20Calculus">Back to top</a> </span></p><p style="line-height: 150%;" align="left"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Image/Sectio9.gif" width="768" border="0" height="10" /><b> <nobr><br /><a name="1.6.2 Some Valid Formulas Over Finite Universe"><br />1.6.2 Some Valid Formulas Over Finite Universe</a><br /></nobr></b><br />We denote predicate formulas by capital letters such as A,B, C,… followed by object variable x, y,… . Thus A(x), A(x, y), B(y) and C(x,y, z) are examples of predicate formulas. In the formula A(x), A is a predicate formula and x is one of the free variables. For example we may write B(x) for (y) P(y) <span style="font-family:Symbol;">Ú</span> Q(x).<br /><br />If in a formula A(x) we replace each free occurrence of the variable x by another variable y, then we say that y is substituted for x in the formula, and the resulting formula is denoted by A(y). For such a substitution, the formula A(x) must be free for y. A formula A(x) is said to be free for y if no free occurrence of x is in the scope of the quantifiers (y) or (<span style="font-family:Symbol;">$</span>y). If A(x) is not free for y, then it is necessary to change the variable y, appearing as a bound variable, to another variable before substituting y for x. If y is substituted, then it is usually a good idea to make all the bound variables different from y. The following examples show what A(y) is for a given A(x).<br /><table width="590" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="35%"><b> </b><p style="line-height: 150%;" align="center"><b>A(x)</b></p></td> <td valign="top" width="65%"><b> </b><p style="line-height: 150%;" align="center"><b>A(y)</b></p></td> </tr> <tr> <td valign="top" width="35%"> <br />P(x,y)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> y)Q(y)</td> <td valign="top" width="65%"> <br />P(y,y)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> y)Q(y) or P(y,y)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> x)Q(z)</td> </tr> <tr> <td valign="top" width="35%"> <br />(S(x)<span style="font-family:Symbol;"> Ù </span>S(y))<span style="font-family:Symbol;">Ú</span>(x)R(x)</td> <td valign="top" width="65%"> <br />(S(y)<span style="font-family:Symbol;"> Ù </span>S(y))<span style="font-family:Symbol;">Ú</span>(x)R(x) or (S(y)<span style="font-family:Symbol;"> Ù </span>S(y))<span style="font-family:Symbol;">Ú</span>(z)R(z)</td> </tr> </tbody></table><br /><br /><br />The following formulas are not free for y P(x,y) <span style="font-family:Symbol;">Ù</span>(y)Q(x,y) and (y)(S(y)<span style="font-family:Symbol;">®</span>S(x)).<br /><br />In order to substitute y in place of the variables x in these formulas it is necessary to first make them free for y as follows.<br /><table width="590" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="50%"><b> </b><p style="line-height: 150%;" align="center"><b>A(x)</b></p></td> <td valign="top" width="50%"><b> </b><p style="line-height: 150%;" align="center"><b>A(y)</b></p></td> </tr> <tr> <td valign="top" width="50%"> <p style="line-height: 150%;" align="center">P(x,y)<span style="font-family:Symbol;"> Ù</span>(z)Q(x,z)</p></td> <td valign="top" width="50%"> <p style="line-height: 150%;" align="center">P(y,y)<span style="font-family:Symbol;">Ù</span>(z)Q(y,z)</p></td> </tr> <tr> <td valign="top" width="50%"> <p style="line-height: 150%;" align="center">(z)(S(z)<span style="font-family:Symbol;">®</span>S(x))</p></td> <td valign="top" width="50%"> <p style="line-height: 150%;" align="center">(z)(S(z)<span style="font-family:Symbol;">®</span>S(y))</p></td> </tr> </tbody></table><br /><br /><br />Let the universe of discourse be denoted by a finite sets given by S={a<sub>1</sub>,a<sub>2</sub>,…,a<sub>n</sub>}.<br /><br /><br /><br />From the meaning of the quantifiers and by simple enumeration of all the objects in S, it is easy to see that<br /><br />(x)A(x)<span style="font-family:Symbol;">Û</span>A(a<sub>1</sub>)<span style="font-family:Symbol;">Ù</span>A(a<sub>2</sub>)<span style="font-family:Symbol;">Ù</span>…<span style="font-family:Symbol;">Ù</span>A(a<sub>n</sub>)…………(1)<br /><br />(<span style="font-family:Symbol;">$</span>x)A(x)<span style="font-family:Symbol;">Û</span>A(a<sub>1</sub>)<span style="font-family:Symbol;">Ú</span>A(a<sub>2</sub>)<span style="font-family:Symbol;">Ú</span>…<span style="font-family:Symbol;">Ú</span>A(a<sub>n</sub>)…………(2)<br /><br />De Morgan's laws are<br /><br /><span style="font-family:Symbol;">ù</span>((x)A(x)) <span style="font-family:Symbol;">Û</span> ( <span style="font-family:Symbol;">$</span>x)<span style="font-family:Symbol;">ù</span>A(x)……………(3)<br /><br /><span style="font-family:Symbol;">ù</span>((<span style="font-family:Symbol;">$</span> x)A(x))<span style="font-family:Symbol;">Û</span>(x)<span style="font-family:Symbol;">ù</span>A(x)……………(4)<br /><br /><br /><span style="font-size:85%;"> </span></p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Section1.6.htm#1.6%20Inference%20Theory%20of%20the%20Predicate%20Calculus">Back to top</a> </span></p><p style="line-height: 150%;" align="left"> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Image/Sectio10.gif" width="778" border="0" height="10" /><b> <nobr><br /><a name="1.6.3 Special Valid Formulas Involving Quantifiers"><br />1.6.3 Special Valid Formulas Involving Quantifiers</a><br /></nobr></b><br />Let A(x) be a predicate formula where x is a particular object variable of interest.<br /><br />Then, (x)A(x)<span style="font-family:Symbol;">Þ</span>A(y)………(1)<br /><br />where y is substituted for x in A(x) to obtain A(y).<br /><br />In order to show (1), we assume that (x)A(x) is true. A(y) must also be true. Hence the implication (1) holds. In case (x)A(x) is false, nothing need to be proved.<br /><br />This implication can be written as (x)A(x)<span style="font-family:Symbol;">Þ</span>A(x)…………(2)<br /><br />Implication (2) will be called the rule of <b>universal specification</b> and will be denoted by <b>US</b> in the theory of inference.<br /><br />Consider A(x)<span style="font-family:Symbol;">Þ</span>(x)A(x)…………(3)<br /><br />This rule is called the rule of <b>universal generalization</b> and is denoted by <b>UG</b>. We also have two more rules which will permit us to remove or add the existential quantifiers during the course of derivation.<br /><br />Consider another two rules<br /><br />(<span style="font-family:Symbol;">$</span>x)A(x)<span style="font-family:Symbol;">Þ</span>A(y)………………(4)<br /><br />A(y)<span style="font-family:Symbol;">Þ</span>(<span style="font-family:Symbol;">$</span>x)A(x)………………(5)<br /><br />Implication (4) is known as <b>existential specification</b>, abbreviated here as <b>ES</b> while (5) is known as <b>existential generalization<i> </i></b>or<b><i> </i>EG</b>.<br /><br />Consider the following tables for the derivation of a conclusion from a set of premises.<br /> <b> <br /> Table1.6.1<br /> </b> <table width="456" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="239"> <p style="line-height: 150%;" align="center">(<span style="font-family:Symbol;">$</span>x)(A(x)<span style="font-family:Symbol;">Ú</span>B(x))<span style="font-family:Symbol;">Û</span>(<span style="font-family:Symbol;">$</span>x)A(x)<span style="font-family:Symbol;">Ú</span>(<span style="font-family:Symbol;">$</span>x)B(x)</p></td> <td valign="top" width="177"> <p style="line-height: 150%;" align="center">E<sub>23</sub></p></td> </tr> <tr> <td valign="top" width="239" height="48"> <p style="line-height: 150%;" align="center">(x)(A(x)<span style="font-family:Symbol;">Ù</span>B(x))<span style="font-family:Symbol;">Û</span>(x)A(x)<span style="font-family:Symbol;">Ù</span>(x)B(x)</p></td> <td valign="top" width="177" height="48"> <p style="line-height: 150%;" align="center">E<sub>24</sub></p></td> </tr> <tr> <td valign="top" width="239" height="36"><span style="font-family:Symbol;"> </span><p style="line-height: 150%;" align="center"><span style="font-family:Symbol;">ù</span>(<span style="font-family:Symbol;">$</span>x)A(x)<span style="font-family:Symbol;">Û</span>(x)<span style="font-family:Symbol;">ù</span>A(x)</p></td> <td valign="top" width="177" height="36"> <p style="line-height: 150%;" align="center">E<sub>25</sub></p></td> </tr> <tr> <td valign="top" width="239"><span style="font-family:Symbol;"> </span><p style="line-height: 150%;" align="center"><span style="font-family:Symbol;">ù</span>(x)A(x)<span style="font-family:Symbol;">Û</span>(<span style="font-family:Symbol;">$</span>x)<span style="font-family:Symbol;">ù</span>A(x)</p></td> <td valign="top" width="177"> <p style="line-height: 150%;" align="center">E<sub>26</sub></p></td> </tr> <tr> <td valign="top" width="239"> <p style="line-height: 150%;" align="center">(x)A(x)<span style="font-family:Symbol;">Ú</span>(x)B(x)<span style="font-family:Symbol;">Þ</span>(x)(A(x)<span style="font-family:Symbol;">Ú</span>B(x))</p></td> <td valign="top" width="177"> <p style="line-height: 150%;" align="center">I<sub>15</sub></p></td> </tr> <tr> <td valign="top" width="239"> <p style="line-height: 150%;" align="center">(<span style="font-family:Symbol;">$</span>x)(A(x)<span style="font-family:Symbol;">Ù</span>B(x))<span style="font-family:Symbol;">Þ</span>(<span style="font-family:Symbol;">$</span>x)A(x)<span style="font-family:Symbol;">Ù</span>(<span style="font-family:Symbol;">$</span>x)B(x)</p></td> <td valign="top" width="177"> <p style="line-height: 150%;" align="center">I<sub>16</sub></p></td> </tr> </tbody></table> <b><br /> Table 1.6.2<br /></b> <table width="356" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="189"> <p style="line-height: 150%;" align="center">(x)(A<span style="font-family:Symbol;">Ú</span>B(x))<span style="font-family:Symbol;">Û</span>A<span style="font-family:Symbol;">Ú</span>(x)B(x)</p></td> <td valign="top" width="127"> <p style="line-height: 150%;" align="center">E<sub>27</sub></p></td> </tr> <tr> <td valign="top" width="189"> <p style="line-height: 150%;" align="center">(<span style="font-family:Symbol;">$</span>x)(A<span style="font-family:Symbol;">Ù</span>B(x))<span style="font-family:Symbol;">Û</span>A<span style="font-family:Symbol;">Ù</span>(<span style="font-family:Symbol;">$</span>x)B(x)</p></td> <td valign="top" width="127"> <p style="line-height: 150%;" align="center">E<sub>28</sub></p></td> </tr> <tr> <td valign="top" width="189"> <p style="line-height: 150%;" align="center">(x)A(x)<span style="font-family:Symbol;">®</span>B<span style="font-family:Symbol;">Û</span>(<span style="font-family:Symbol;">$</span>x)(A(x)<span style="font-family:Symbol;">®</span> B)</p></td> <td valign="top" width="127"> <p style="line-height: 150%;" align="center">E<sub>29</sub></p></td> </tr> <tr> <td valign="top" width="189"> <p style="line-height: 150%;" align="center">(<span style="font-family:Symbol;">$</span>x)A(x)<span style="font-family:Symbol;">®</span>B<span style="font-family:Symbol;">Û</span>(x)(A(x)<span style="font-family:Symbol;">®</span> B)</p></td> <td valign="top" width="127"> <p style="line-height: 150%;" align="center">E<sub>30</sub></p></td> </tr> <tr> <td valign="top" width="189"> <p style="line-height: 150%;" align="center">A<span style="font-family:Symbol;">®</span>(x)B(x)<span style="font-family:Symbol;">Û</span>(x)(A<span style="font-family:Symbol;">®</span>B(x))</p></td> <td valign="top" width="127"> <p style="line-height: 150%;" align="center">E<sub>31</sub></p></td> </tr> <tr> <td valign="top" width="189"> <p style="line-height: 150%;" align="center">A<span style="font-family:Symbol;">®</span>(<span style="font-family:Symbol;">$</span>x)B(x)<span style="font-family:Symbol;">Û</span>(<span style="font-family:Symbol;">$</span>x)(A<span style="font-family:Symbol;">®</span>B(x))</p></td> <td valign="top" width="127"> <p style="line-height: 150%;" align="center">E<sub>32</sub></p></td> </tr> </tbody></table> </p><p style="line-height: 150%;"><br /></p><p style="line-height: 150%;" align="right"> <span style="font-size:85%;"> <a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Section1.6.htm#1.6%20Inference%20Theory%20of%20the%20Predicate%20Calculus">Back to top</a> </span> </p><p style="line-height: 150%;"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Image/Sectio11.gif" width="780" border="0" height="10" /><b> <nobr><br /><a name="1.6.4 Theory of Inference for the Predicate Calculus"><br />1.6.4 Theory of Inference for the Predicate Calculus</a><br /></nobr></b><br />The method of derivation involving predicate formulas uses the rules of inference given for the statement calculus and also certain additional rules which are required to deal with the formulas involving quantifiers. Using of rules P and T remain the same. If the conclusion is given in the form of a conditional, we shall use the rule of conditional proof called <b>CP</b>. In order to use the equivalences and implications, we need some rules on how to eliminate quantifiers during the course of derivation. This elimination is done by the rules of specification, called rules US and ES. Once the quantifiers are eliminated and the conclusion is reached. It may happen that the desired conclusion is quantified. In this case we need rules of generalization called rules UG and EG which can be used to attach a quantifier.<br /><b><br />Rule US (Universal Specification) :<br /></b><br />From (x)A(x) one can conclude A(y).<br /><br /><b>Rule ES (Existential Specification) :<br /></b><br />From (<span style="font-family:Symbol;">$</span>x)A(x) one can conclude A(y) provided that y is not free in any given premise and also not free in any prior step of the derivation. These requirements can easily be met by choosing a new variable each time ES is used.<br /><br /><b>Rule EG (Existential Generalization) :<br /></b><br />From A(x) one can conclude (<span style="font-family:Symbol;">$</span> y)A(y).<br /><b><br />Rule UG (Universal Generalization) :<br /></b><br />From A(x) one can conclude (y)A(y) provided that x is not free in any of the given premises and provided that if x is free in a prior step which resulted from use of ES, then no variables introduced by that use of ES appear free in A(x).<br /><b><br />Example1:<br /></b><br />Show that (x) (H(x)<span style="font-family:Symbol;">®</span> M(x))<span style="font-family:Symbol;"> Ù </span>H(s)<span style="font-family:Symbol;">Þ</span> M(s).<br /><br />This is a well - known argument known as the "Socrates argument" which is given by<br /><br />All men are mortal.<br /><br />Socrates is a man.<br /><br />Therefore Socrates is mortal.<br /><br />If we denote H(x) : x is a man, M(x) : x is a mortal and s : Socrates, we can put the argument in the above form.<br /><b><br />Solution:<br /></b><br />{1} (1) (x)(H(x)<span style="font-family:Symbol;">®</span> M(x)) Rule P<br /><br />{1} (2) H(s)<span style="font-family:Symbol;">®</span> M(s) Rule US, (1)<br /><br />{3} (3) H(s) Rule P<br /><br />{1, 3} (4) M(s) Rule T, (2), (3), I<sub>11<br /></sub> <b><br /><br /><br />Example 2:<br /></b><br />Show that (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)) <span style="font-family:Symbol;">Ù</span> (x)(Q(x)<span style="font-family:Symbol;">®</span> R(x)) <span style="font-family:Symbol;">Þ</span> (x)(P(x)<span style="font-family:Symbol;">®</span> R(x)).<br /><b><br />Solution:<br /></b><br />{1} (1) (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)) Rule P<br /><br />{1} (2) P(y)<span style="font-family:Symbol;">®</span> Q(y) Rule US, (1)<br /><br />{3} (3) (x)(Q(x)<span style="font-family:Symbol;">®</span> R(x)) Rule P<br /><br />{4} (4) Q(y)<span style="font-family:Symbol;">®</span> R(y) Rule US, (3)<br /><br />{1,3} (5) P(y)<span style="font-family:Symbol;">®</span> R(y) Rule T, (2), (4), I<sub>13<br /></sub><br />{1, 3} (6) (x)(P(x)<span style="font-family:Symbol;">®</span> R(x)) Rule UG, (5)<br /><b> <br /><br /> <br />Example 3:<br /></b> <br />Show that (<span style="font-family:Symbol;">$</span> x)M(x) follows logically from the premises (x)(H(x)<span style="font-family:Symbol;">®</span> M(x)) and (<span style="font-family:Symbol;">$</span> x)H(x).<br /> <b> <br />Solution:<br /></b><br />{1} (1) (x)(H(x)<span style="font-family:Symbol;">®</span> M(x)) Rule P<br /><br />{1} (2) H(y)<span style="font-family:Symbol;">®</span> M(y) Rule US, (1)<br /><br />{3} (3) (<span style="font-family:Symbol;">$</span> x)H(x) Rule P<br /><br />{3} (4) H(y) Rule ES, (3)<br /><br />{1, 3} (5) M(y) Rule T, (2), (4), I<sub>11<br /></sub><br />{1, 3} (6) (<span style="font-family:Symbol;">$</span> x)M(x) Rule EG, (5)<br /> <b><br /><br /><br />Example 4:<br /></b><br />Prove that (<span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>Q(x)) <span style="font-family:Symbol;">Þ</span> (<span style="font-family:Symbol;">$</span> x)P(x) <span style="font-family:Symbol;">Ù</span> (<span style="font-family:Symbol;">$</span> x)Q(x).<br /><b><br />Solution:<br /></b><br />{1} (1) (<span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;">Ù</span> Q(x)) Rule P<br /><br />{1} (2) P(y) <span style="font-family:Symbol;">Ù</span> Q(y) Rule ES, (1), y fixed<br /><br />{1} (3) P(y) Rule T, (2), I<sub>1<br /></sub><br />{1} (4) Q(y) Rule T, (2), I<sub>2<br /></sub><br />{1} (5) (<span style="font-family:Symbol;">$</span> x)P(x) Rule EG, (3)<br /><br />{1} (6) (<span style="font-family:Symbol;">$</span> x)Q(x) Rule EG, (4)<br /><br />{1} (7) (<span style="font-family:Symbol;">$</span> x)P(x)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> x)Q(x) Rule T, (4), (5), I<sub>9<br /></sub> <b><br /><br /><br />Example 5:<br /></b><br />Show that from<br /></p><ol type="a"><ol type="a"><li> (<span style="font-family:Symbol;">$</span> x)(F(x)<span style="font-family:Symbol;"> Ù </span>S(x)) <span style="font-family:Symbol;">®</span> (y)(M(y)<span style="font-family:Symbol;">®</span> W(y))</li><li> (<span style="font-family:Symbol;">$</span> y)(M(y)<span style="font-family:Symbol;"> Ù ù</span>W(y))</li></ol></ol> the conclusion (x)(F(X) <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>S(x)) follows.<br /><b><br />Solution:<br /></b><br />{1} (1) (<span style="font-family:Symbol;">$</span> y)(M(y)<span style="font-family:Symbol;"> Ù ù</span>W(y)) Rule P<br /><br />{1} (2) M(z)<span style="font-family:Symbol;"> Ù ù</span>W(z) Rule ES, (1)<br /> <br />{1} (3) <span style="font-family:Symbol;">ù</span>(M(z)<span style="font-family:Symbol;">®</span> W(z)) Rule T, (2), E<sub>17<br /></sub><br />{1} (4) (<span style="font-family:Symbol;">$</span> y)<span style="font-family:Symbol;">ù</span>(M(y)<span style="font-family:Symbol;">®</span> W(y)) Rule EG, (3)<br /><br />{1} (5) <span style="font-family:Symbol;"> ù</span>(y)(M(y)<span style="font-family:Symbol;">®</span> W(y)) E<sub>26</sub>, (4)<br /><br />{6} (6) (<span style="font-family:Symbol;">$</span> x)(F(x) <span style="font-family:Symbol;">Ù</span> S(x))<span style="font-family:Symbol;">®</span> (y)(M(y)<span style="font-family:Symbol;">®</span> W(y)) Rule P<br /><br />{1, 6} (7) <span style="font-family:Symbol;"> ù</span>(<span style="font-family:Symbol;">$</span> x)(F(x) <span style="font-family:Symbol;">Ù</span> S(x)) Rule T, (5), (6), I<sub>12<br /></sub><br />{1, 6} (8) (x)<span style="font-family:Symbol;">ù</span>(F(x) <span style="font-family:Symbol;">Ù</span> S(x)) Rule T,(7), E<sub>25<br /></sub><br />{1, 6} (9) <span style="font-family:Symbol;"> ù</span>(F(x) <span style="font-family:Symbol;">Ù</span> S(x)) Rule US, (8)<br /><br />{1, 6} (10) F(x)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>S(x) Rule T, (9), E<sub>9</sub>, E<sub>16</sub>, E<sub>17<br /></sub><br />{1, 6} (11) (x)(F(x)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>S(x)) Rule UG, (10)<br /><b><br /><br /><br />Example 6:<br /></b><br />Show that (x)(P(x) <span style="font-family:Symbol;">Ú </span>Q(x)) <span style="font-family:Symbol;">Þ</span> (x)P(x) <span style="font-family:Symbol;">Ú </span>(<span style="font-family:Symbol;">$</span> x)Q(x).<br /><b><br />Solution:<br /></b><br />We shall use the indirect method of proof by assuming <span style="font-family:Symbol;">ù</span>((x)P(x)<span style="font-family:Symbol;">Ú</span> (<span style="font-family:Symbol;">$</span> x)Q(x)) as an additional premises.<br /><br />{1} (1) <span style="font-family:Symbol;">ù</span>((x)P(x)<span style="font-family:Symbol;">Ú</span>( <span style="font-family:Symbol;">$</span> x)Q(x)) Rule P(assumed)<br /><br />{1} (2) <span style="font-family:Symbol;">ù</span>(x)P(x)<span style="font-family:Symbol;"> Ù ù</span>(<span style="font-family:Symbol;">$</span> x)Q(x) Rule T, (1), E<sub>9<br /></sub><br />{1} (3) <span style="font-family:Symbol;">ù</span>(x)P(x) Rule T, (2), I<sub>1<br /></sub><br />{1} (4) (<span style="font-family:Symbol;">$</span> x)<span style="font-family:Symbol;">ù</span>P(x) Rule T, (3), E<sub>26<br /></sub><br />{1} (5) <span style="font-family:Symbol;">ù</span>(<span style="font-family:Symbol;">$</span> x)Q(x) Rule T, (2), I<sub>2<br /></sub><br />{1} (6) (x)<span style="font-family:Symbol;">ù</span>Q(x) Rule T, (5), E<sub>25<br /></sub><br />{1} (7) <span style="font-family:Symbol;"> ù</span>P(y) Rule ES, (4)<br /><br />{1} (8) <span style="font-family:Symbol;">ù</span>Q(y) Rule US, (6)<br /><br />{1} (9) <span style="font-family:Symbol;">ù</span>P(y)<span style="font-family:Symbol;"> Ù ù</span>Q(y) Rule T, (7), (8), I<sub>9<br /></sub><br />{1} (10) <span style="font-family:Symbol;">ù</span>(P(y)<span style="font-family:Symbol;">Ú</span>Q(y)) Rule T, (9), E<sub>9<br /></sub><br />{11} (11) (x)(P(x)<span style="font-family:Symbol;">Ú</span>Q(x)) Rule P<br /><br />{11} (12) <span style="font-family:Symbol;">ù</span>(P(y)<span style="font-family:Symbol;">Ú</span>Q(y))<span style="font-family:Symbol;"> Ù </span>(P(y)<span style="font-family:Symbol;">Ú</span> Q(y)) Rule T, (10), (12), I<sub>9<br /> </sub> <br /> Contradiction.<br /><br /><br /> <p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Section1.6.htm#1.6%20Inference%20Theory%20of%20the%20Predicate%20Calculus">Back to top</a></span> </p><p style="line-height: 150%;" align="left"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.6/Image/Sectio12.gif" width="772" border="0" height="10" /><br /> <b> <nobr><br /><span style="font-size:130%;"><a name="Exercise :">Exercise:</a></span><br /></nobr></b> </p><ol><li> <p style="line-height: 150%;">Show that P(x)<span style="font-family:Symbol;"> Ù (</span>x)Q(x)<span style="font-family:Symbol;">Þ</span> ( <span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>Q(x)). </p></li><li> Explain why the following steps in the derivations are not correct</li></ol> <blockquote> <blockquote> (a) (1) (x)P(x)<span style="font-family:Symbol;">®</span> Q(x)<br /> <br /> (2) P(x)<span style="font-family:Symbol;">®</span> Q(x) (1), Rule US<br /> <br />(b) (1) (x)P(x)<span style="font-family:Symbol;">®</span> Q(x)<br /> <br /> (2) P(y)<span style="font-family:Symbol;">®</span> Q(x) (1), Rule US<br /> <br />(c) (1) (x)(P(x)<span style="font-family:Symbol;">Ú</span>Q(x))<br /> <br /> (2) P(a)<span style="font-family:Symbol;">Ú</span>Q(b) (1), Rule US<br /> <br />(d) (1) (x)(P(x)<span style="font-family:Symbol;">Ú</span>( <span style="font-family:Symbol;">$</span> x)(Q(x)<span style="font-family:Symbol;"> Ù </span>R(x)))<br /> <br /> (2) P(a)<span style="font-family:Symbol;">Ú</span>( <span style="font-family:Symbol;">$</span> x)(Q(x)<span style="font-family:Symbol;"> Ù </span>R(a)) (1), Rule US<br /> </blockquote> </blockquote> <br /> 3. What is wrong in the following steps of derivation?<br /><br /> (a) (1) P(x)<span style="font-family:Symbol;">®</span> Q(x) Rule P<br /><br /> (2) (<span style="font-family:Symbol;">$</span> x)P(x)<span style="font-family:Symbol;">®</span> Q(x) (1), Rule EG<br /><br /> (b) (1) P(a)<span style="font-family:Symbol;">®</span> Q(b) Rule P<br /><br /> (2) (<span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;">®</span> Q(x) (1), Rule EG<br /><br /> (c) (1) P(a)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> x)(P(a)<span style="font-family:Symbol;"> Ù </span>Q(x)) Rule P<br /><br /> (2) (<span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>Q(x))) (1), Rule EG<br /><br /> 4. Demonstrate the following implications.<br /><ol type="a"><ol type="a"><li> <span style="font-family:Symbol;">ù</span>((<span style="font-family:Symbol;">$</span> x)P(x)<span style="font-family:Symbol;"> Ù </span>Q(a))<span style="font-family:Symbol;">Þ</span> ( <span style="font-family:Symbol;">$</span> x)P(x)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>Q(a).</li><li> (x)(<span style="font-family:Symbol;">ù</span>P(x)<span style="font-family:Symbol;">®</span> Q(x)), (x)<span style="font-family:Symbol;">ù</span>Q(x)<span style="font-family:Symbol;">Þ</span> P(a).</li><li> (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)),(x)(Q(x)<span style="font-family:Symbol;">®</span> R(x))<span style="font-family:Symbol;">Þ</span> P(x)<span style="font-family:Symbol;">®</span> R(x).</li><li> (x)(P(x)<span style="font-family:Symbol;">Ú</span>Q(x)), (x)<span style="font-family:Symbol;">ù</span>P(x)<span style="font-family:Symbol;">Þ</span> ( <span style="font-family:Symbol;">$</span> x) Q(x).</li><li> (x)(P(x)<span style="font-family:Symbol;">Ú</span>Q(x), (x)<span style="font-family:Symbol;">ù</span>P(x)<span style="font-family:Symbol;">Þ</span> (x)Q(x).</li><li> <span style="font-family:Symbol;">ù</span>(x)(P(x)<span style="font-family:Symbol;"> Ù </span>Q(x)), (x)P(x)<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span>(x)Q(x).</li></ol></ol> <br /> 5. There is a mistake in the following derivation. Find it. Is the conclusion valid? If so, obtain a correct<br /> derivation.<br /> <br /> {1} (1) (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)) Rule P<br /><br /> {1} (2) P(y)<span style="font-family:Symbol;">®</span> Q(y) Rule US, (1)<br /><br /> {3} (3) (<span style="font-family:Symbol;">$</span> x)P(x) Rule P<br /><br /> {3} (4) P(y) Rule ES, (3)<br /><br /> {1,3} (5) Q(y) Rule T, (2), (4), I<sub>11<br /></sub><br /> {1,3} (6) (<span style="font-family:Symbol;">$</span>x)Q(x) Rule EG, (5)<br /><br /> 6. Are the following conclusions validly derivable from the premises given?<br /><br /> (a) (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)), (<span style="font-family:Symbol;">$</span> y)P(y) <b>C</b> : ( <span style="font-family:Symbol;">$</span> z)Q(z)<br /><br /> (b) (<span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>Q(x)) <b>C </b>: (x)P(x)<br /><br /> (c) (<span style="font-family:Symbol;">$</span> x)P(x), (<span style="font-family:Symbol;">$</span> x)Q(x) <b>C </b>: ( <span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>Q(x))<br /><br /> (d) (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)), <span style="font-family:Symbol;">ù</span>Q(a) <b>C </b>: (x)<span style="font-family:Symbol;">ù</span>P(x)<br /><br /> 7. Using CP or otherwise, show the following implications<br /> (a) (<span style="font-family:Symbol;">$</span> x)P(x)<span style="font-family:Symbol;">®</span> (x)Q(x)<span style="font-family:Symbol;">Þ</span> (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)).<br /><br /> (b) (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)), (x)(R(x)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>Q(x)) <span style="font-family:Symbol;">Þ</span> (x)(R(x)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>P(x)).<br /><br /> (c) (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)) <span style="font-family:Symbol;">Þ</span> (x)P(x)<span style="font-family:Symbol;">®</span> (x)Q(x).<br /><br /> 8. Show the following by constructing derivations<br /><br /> (a) (<span style="font-family:Symbol;">$</span> x)P(x)<span style="font-family:Symbol;">®</span> (x)((P(x)<span style="font-family:Symbol;">Ú</span> Q(x))<span style="font-family:Symbol;">®</span> R(x)), (<span style="font-family:Symbol;">$</span> x)P(x),(<span style="font-family:Symbol;">$</span> x)Q(x)<span style="font-family:Symbol;">Þ</span> (<span style="font-family:Symbol;">$</span> x)( <span style="font-family:Symbol;">$</span> y)(R(x)<span style="font-family:Symbol;"> Ù </span>R(y)).<br /><br /> (b) (x)(P(x)<span style="font-family:Symbol;">®</span> (Q(y)<span style="font-family:Symbol;"> Ù </span>R(x))), (<span style="font-family:Symbol;">$</span> x)P(x) <span style="font-family:Symbol;">Þ</span> Q(y)<span style="font-family:Symbol;"> Ù </span>( <span style="font-family:Symbol;">$</span> x)(P(x)<span style="font-family:Symbol;"> Ù </span>R(x)).Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1tag:blogger.com,1999:blog-7399621945608861143.post-60708415015444978622008-08-12T04:21:00.004-07:002008-12-23T01:55:02.555-08:00Predicate Calculus<p align="left"><nobr> <b><a name="Predicate Calculus"><br /></a></b><br />It is not possible to express the fact that any two atomic statements have some features in common. In order to investigate questions of the nature, we introduce the concept of a predicate in an atomic statement. <i>The logic based upon the analysis of predicates in any statement is called <b>predicate logic</b>.<br /></i> <b><a name="1.5.1 Predicates"><br /><br />1.5.1 Predicates<br /><br /></a></b>Consider two statements<br /> <ol><li> <br />John is a bachelor</li><li> <br />Smith is a bachelor</li></ol> <br />The part ‘is a bachelor’ is called a predicate<br /><br />Denote John by j and Smith by s, ‘is a bachelor’ by predicate letter B. The statements (1) and (2) may be written as B(j) and B(s).<br /><br />In general "p is Q", where Q is predicate and p is the subject, can be denoted by Q(p). A statement, which is expressed by using a predicate letter must have atleast one name of an object associated with the predicate.<br /><br />3. John is a bachelor, and this painting is red.<br /><br />Let R denote the predicate "is red" and let p denote "this painting". Then the statement (3) can be written as B(j) <span style="font-family:Symbol;">Ù</span> R(p). Other connectives can also be used to form statements such as B(j)<span style="font-family:Symbol;">®</span>R(p), <span style="font-family:Symbol;">ù</span>R(p), B(j)<span style="font-family:Symbol;">Ú</span>R(p), etc.<br /><br />Consider, now, statements involving the names of two objects, such as<br /><br />4. James is taller than Jones<br /><br />5. China is to the north of India<br /><br />The predicates "is taller than" and "is to the north of" are 2-place predicates because names of two objects are needed to complete a statement involving these predicates. If the letter G symbolizes "is taller than", j<sub>1</sub> denotes "James", and j<sub>2</sub> denotes "Jones", then statement (4) can be translated as G(j<sub>1</sub>, j<sub>2</sub>). Note that the order in which the names appear in the statement as well as in the predicate is important. If N denotes the predicate "is to the north of", c : China, and i : India, then (5) is symbolized as N(c, i). N(c, i) is the statement "India is to the north of China".<br /><br />6. Suji stands between Rani and Ravi.<br /><br />7. Siva and Lazar played bridge against Jacob and Senthil.<br /><br />If S is an n-place predicate letter and a<sub>1</sub>,a<sub>2</sub>,…,a<sub>n</sub> are the names of objects, then S(a<sub>1</sub>,a<sub>2</sub>,…,a<sub>n</sub>) is a statement.<br /></nobr><br /><br /><span style="font-size:85%;"> </span></p><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Section1.5.htm#1.5%20The%20Predicate%20Calculus">Back to top</a> </span></p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Image/Sectio3.gif" width="766" border="0" height="10" /><br /><nobr><br /><a name="1.5.2 The Statement Function, Variables and Quantifiers">1.5.2 The Statement Function, Variables and Quantifiers</a><br /></nobr></b><br />Let H be the predicate "is a mortal", a : the name "Mr.Suji", b : India" and c : "A Shirt".<br /><br />Then H(a), H(b) and H(c) all denote statements. These statements have a common form.<br /><br />If we write H(x) for "x is mortal", then H(a), H(b), H(c), and others having the same form can be obtained from H(x) by replacing x by an appropriate name. H(x) is not a statement but it results in a statement when x is replaced by the name of an object.<br /><br />A simple statement function of one variable is defined to be an expression consisting of a predicate symbol and an individual variable. This statement function becomes statement when the variable is replaced by any object. The replacement is called a substitution instance of the statement function.<br /><br />For example, if we let M(x) be "x is a man" and H(x) be "x is a mortal", then we can form compound statement functions such as<br /><br />M(x)<span style="font-family:Symbol;">Ù</span> H(x), M(x)<span style="font-family:Symbol;">®</span> H(x), <span style="font-family:Symbol;">ù</span>H(x), M(x)<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span>H(x), etc.<br /><br />Consider the statement function of two variables :<br /></p><ol><li> H(x, y) : x is taller than y.</li></ol> If both x and y are replaced by the name of objects we get a statement. If i represents Ms. Indu and j Mr. Jeyaraj, then we have,<br /> <br /> H(i, j) : Ms. Indu is taller than Mr. Jeyaraj and<br /> <br /> H(j, i) : Mr. Jeyaraj is taller than Ms. Indu.<br /> <br /><br /> <br />It is possible to form statement functions of two variables by using statement functions of one variable<br /> <br />For example,<br /> <br />M(x) : x is a man<br /> <br />H(y): y is mortal.<br /> <br />Then we may write<br /> <br />M(x) <span style="font-family:Symbol;">Ù</span> H(y) : x is a man and y is a mortal.<br /> <br />It is not possible to write every statement function of two variables using statement functions of one variable consider equation in elementary algebra.<br /><ol><li> x + 4 = 10.</li><li> x<sup>2 </sup>+ 2x + 5 = 0.</li><li> (x – 5) * ( x – ¼) = 0.</li><li> (x<sup>2 </sup>– 1) = (x + 1) * (x – 1).</li></ol> In algebra, it is conventional to assume that the variable x is to be replaced by numbers (real, complex, rational, integer, etc.). We state this idea by saying that the universe of the variable x is the set of real numbers or complex numbers or integers etc.<br /><br />If x is replaced by a real number in (1), we get a statement. The resulting statement is true when 6 is substituted for x, while, for every other substitution, the resulting statement is false.<br /><br />In (2) there is no real number which when substituted for x gives a true statement. In (3) if the universe of x is assumed to be integers, then there is only one number which produces a true statement when substituted. In (4), if any number is substituted for x, then the resulting statement is true.<br /><br />Therefore we may say that<br /><br /> 5. For a number x, x<sup>2</sup> – 1 = ( x – 1 ) * ( x + 1 ).<br /><br />Here (5) is a statement and not a statement function even though the variable x appears in it. In (5) the variable x need not be replaced by any name to obtain a statement.<br /><br />Consider the following statements. Each one is a statement about all individuals or objects belonging to a certain set.<br /><br /> 6. All men are mortal.<br /> 7. Every apple is red.<br /> 8. Any integer is either positive or negative.<br /><br />The above statements may be written as follows:<br /><br /> 6a. For all x, if x is a man, then x is mortal.<br /><br /> 7a. For all x, if x is an apple, then x is red.<br /><br /> 8a. For all x, if x is an integer, then x is either positive or negative.<br /><br /><br />If we introduce a symbol to denote the phrase "for all x" by the symbol "(<span style="font-family:Symbol;">"</span> x)" or by "(x)", then we may write as<br /><br />M(x) : x is a man. H(x) : x is mortal.<br /><br />A(x) : x is an apple. R(x) : x is red.<br /><br />N(x) : x is an integer. P(x) : x is either positive or negative.<br /><br />Using the above we write (6a), (7a) and (8a) as<br /><br /> 6b. (x)(M(x)<span style="font-family:Symbol;">®</span> H(x)) or (<span style="font-family:Symbol;">"</span> x)(M(x)<span style="font-family:Symbol;">®</span> H(x)).<br /><br /> 7b. (x)(A(x)<span style="font-family:Symbol;">®</span> R(x)) or (<span style="font-family:Symbol;">"</span> x)(A(x)<span style="font-family:Symbol;">®</span> R(x)).<br /><br /> 8b. (x)(N(x)<span style="font-family:Symbol;">®</span> P(x)) or (<span style="font-family:Symbol;">"</span> x)(N(x)<span style="font-family:Symbol;">®</span> P(x)).<br /><br /><br /><br />The symbols (x) or (<span style="font-family:Symbol;">"</span> x) are called universal quantifiers. It is possible to quantify any statement function of one variable to obtain a statement. (x)M(x) is a statement which can be translated as<br /><br /> 9. For all x, x is a man.<br /><br /> 9a. For every x, x is a man.<br /><br /> 9b. Every thing is a man.<br /><br /><br />The statements remain unchanged if x is replaced by y through out.<br /><br />Therefore the statements (x)(M(x)<span style="font-family:Symbol;">®</span> H(x)) and (y)(M(y)<span style="font-family:Symbol;">®</span> H(y)) are equivalent.<br /><br /><br />Consider an example of two universal quantifiers.<br /><br /> H(x, y) : x is taller than y.<br /><br />We can state that "For any x and any y, if x is taller than y, then y is not taller than x" or "For any x and y, if x is taller than y, then it is not true that y is taller than x".<br /><br />This may be symbolized as<br /><br />(x)(y)(H(x, y)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>H(y, x))<br /><br /><br />Another quantifier (<span style="font-family:Symbol;">$</span> x) is used to translate the expression such as "for some", "there is atleast one", or "there exists some" ("some" is used in the sense of "atleast one").<br /><br />Consider the following statements :<br /><br /> 10. There exists a man.<br /><br /> 11. Some men are clever.<br /><br /> 12. Some real numbers are rational.<br /><br /><br />(10) can be written as<br /><br /> 10a. There exists an x such that x is a man.<br /><br /> 10b. There is atleast one x such that x is a man.<br /><br />(11) can be written as<br /><br /> 11a. There exists x such that x is a man and x is clever.<br /><br /> 11b. There exists atleast one x such that x is a man and x is clever.<br /><br /><br />The phrase "there is atleast one x such that" or "there exists x such that" or "for some x" is represented by the symbol "<span style="font-family:Symbol;">$</span> x" called the existential quantifiers. Writing<br /><br /> M(x) : x is a man.<br /><br /> C(x) : x is clever.<br /><br /> R<sub>1</sub>(x) : x is a real number.<br /><br /> R<sub>2</sub>(x) : x is rational.<br /><br />We can write (10) to (12) as<br /><br /> 10c. (<span style="font-family:Symbol;">$</span> x) M(x)<br /><br /> 11c. (<span style="font-family:Symbol;">$</span> x)(M(x)<span style="font-family:Symbol;">Ù</span> C(x))<br /><br /> 12c. (<span style="font-family:Symbol;">$</span> x)(R<sub>1</sub>(x)<span style="font-family:Symbol;">Ù</span> R<sub>2</sub>(x))<br /><b><br /><br /></b><span style="font-size:85%;"> </span><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Section1.5.htm#1.5%20The%20Predicate%20Calculus">Back to top</a> </span></p><p style="line-height: 150%;" align="left"> <b> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Image/Sectio4.gif" width="775" border="0" height="10" /> <nobr><br /><a name="1.5.3 Predicate Formulas">1.5.3 Predicate Formulas</a><br /></nobr></b><br />Consider n-place predicate formulas P(x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>)<br /><br />The letter P is an n-place predicate and x<sub>1</sub>,x<sub>2</sub>, … , x<sub>n</sub> are individual variables. In general P(x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>) is called an atomic formula of predicate calculus. Some examples of atomic formulas:<br /><br />P, Q(x), R(x, y), A(x, y, z), B(a, y), C(x, a, z).<br /><br /><br /><i> A well-formed formula of predicate calculus is obtained by using the following rules.<br /></i></p><ol><i> </i><ol><i> </i><li> <i> An atomic formula is a well-formed formula.</i></li><i> </i><li> <i> If A is a wff, then <span style="font-family:Symbol;">ù</span></i> <span style="font-family:Symbol;"> </span> <i> A is a wff.</i></li><i> </i><li> <i> If A and B are wffs, then (A</i><span style="font-family:Symbol;">Ù</span> <i> B), (A</i><span style="font-family:Symbol;">Ú</span> <i> B), (A</i><span style="font-family:Symbol;">®</span> <i> B), and (A</i><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> <i> B) are also wffs.</i></li><i> </i><li> <i> If A is a wff and x is any variable, then (x)A and (<span style="font-family:Symbol;">$</span> x)A are wffs.</i></li><i> </i><li> <i> Only those formulas obtained by using rules (1) to (4) are wffs.</i></li><i> </i></ol></ol> <span style="font-size:85%;"> </span><p style="line-height: 150%;" align="right"><span style="font-size:85%;"> <a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Section1.5.htm#1.5%20The%20Predicate%20Calculus">Back to top</a> </span></p><p style="line-height: 150%;" align="left"><b><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Image/Sectio5.gif" width="771" border="0" height="10" /> <nobr><br /><a name="1.5.4 Free and Bound Variables"><br />1.5.4 Free and Bound Variables</a></nobr></b><br /><i>Given a formula containing a part of the form (x)P(x) or (<span style="font-family:Symbol;">$</span> x)P(x), such a part is called an x-bound part of the formula. Any occurrence of x in an x-bound part of a formula is called a <b>bound occurrence</b> of x, while any occurrence of x, or of any variable, that is not bound is called a <b>free occurrence</b></i>.<br />The formula P(x) either in (x) P(x) or in (<span style="font-family:Symbol;">$</span> x)P(x) is described as the scope of the quantifier. If the scope is an atomic formula, then no parentheses are used to enclose the formula.<br /><br />Consider the following formulas as examples :<br /></p><ol><ol><li> (x)P(x, y).</li><li> (x) (P(x)<span style="font-family:Symbol;">®</span> Q(x)).</li><li> (x)(P(x)<span style="font-family:Symbol;">®</span> (<span style="font-family:Symbol;">$</span>y)R(x, y)).</li><li> (x)(P(x)<span style="font-family:Symbol;">®</span> R(x)) <span style="font-family:Symbol;">Ú</span> (x)(P(x)<span style="font-family:Symbol;">®</span> Q(x)).</li><li> (<span style="font-family:Symbol;">$</span>x)(P(x)<span style="font-family:Symbol;">Ù</span> Q(x)).</li><li> (<span style="font-family:Symbol;">$</span>x)P(x) <span style="font-family:Symbol;">Ù</span> Q(x).</li></ol></ol> In (1), P(x, y) is the scope of the quantifiers, and both occurrences of x are bound occurrences, while the occurrence of y is a free occurrence. In (2), the scope of the universal quantifier is P(x)<span style="font-family:Symbol;">®</span>Q(x), and all occurrences of x are bound. In (3), the scope of (x) is P(x)<span style="font-family:Symbol;">®</span>(<span style="font-family:Symbol;">$</span>y)R(x, y), while the scope of (<span style="font-family:Symbol;">$</span>y) is R(x, y). All occurrences of both x and y are bound occurrences. In (4), the scope of the first quantifier is P(x)<span style="font-family:Symbol;">®</span>R(x), and the scope of the second is P(x)<span style="font-family:Symbol;">®</span>Q(x). All occurrences of x are bound occurrences. In (5), the scope of (<span style="font-family:Symbol;">$</span>x) is P(x)<span style="font-family:Symbol;">Ù</span>Q(x). In (6), the scope of (<span style="font-family:Symbol;">$</span>x) is P(x), and the last occurrence of x in Q(x) is free.<br /><br />We may write (x)P(x, y) as (z)P(z, y). The bound occurrence of a variable can not be substituted by a constant. Only a free occurrence of a variable can be substituted by a constant.<br /><br />For example, (x)P(x)<span style="font-family:Symbol;">Ù</span>Q(a) is a substitution instance of (x)P(x)<span style="font-family:Symbol;">Ù</span>Q(y). (x)P(x)<span style="font-family:Symbol;">Ù</span>Q(a) can be expressed in English as "Every x has the property P, and ‘a’ has the property Q". A change of variables in the bound occurrence is not a substitution instance. In (6), it is better to write (y)P(y)<span style="font-family:Symbol;">Ù</span>Q(x) instead of (x)P(x)<span style="font-family:Symbol;">Ù</span>Q(x), so as to separate the free and bound occurrences of variables. It may be mentioned that in a statement every occurrence of a variable must be bound, and no variable should have a free occurrence.<br /><b><br /><br />Example 1:<br /></b><br />Let P(x) : x is person.<br /><br />F(x, y) : x is the father of y.<br /><br />M(x, y) : x is the mother of y.<br /><br />Write the predicate " x is the father of the mother of y".<br /><b><br />Solution:<br /></b><br />In order to symbolize the predicate, we name a person called z as the mother of y. That is we want to say that x is the father of z and z the mother of y. It is assumed that such a person z exists.<br /><br />We symbolize the predicate as (<span style="font-family:Symbol;">$</span>z)(P(z) <span style="font-family:Symbol;">Ù</span> F(x, z) <span style="font-family:Symbol;">Ù</span> M(z, y)).<br /><br /><br /><b><br />Example 2:<br /></b><br />Symbolize the expression. "All the world loves a lover".<br /><b><br />Solution:<br /></b><br />It means that everybody loves a lover.<br /><br />Let P(x): x is person.<br /><br />L(x): x is a lover.<br /><br />R(x. y): x loves y.<br /><br />The required expression is (x)(P(x) <span style="font-family:Symbol;">® </span>(y)(P(y) <span style="font-family:Symbol;">Ù </span>L(y) <span style="font-family:Symbol;">® </span>R(x,y))).<br /><br /><br /><span style="font-size:85%;"> </span><p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Section1.5.htm#1.5%20The%20Predicate%20Calculus">Back to top</a> </span></p><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.5/Image/Sectio6.gif" width="761" border="0" height="10" /><b> <nobr><br /><a name="1.5.5 The Universe Of Discourse"><br />1.5.5 The Universe Of Discourse</a><br /></nobr></b><br />When we symbolize a statement some simplification can be introduced by limiting the class of individuals or objects. The limitation means that the variables which are quantified stand for only those objects which are members of a particular set or class. Such a restricted class is called the universe of discourse or the domain of individuals or simply the universe. If it refers to human beings only, then the universe of discourse is the class of human beings.<br /><b><br />Example 1:<br /></b><br />Symbolize the statement "All men are giants".<br /><b><br />Solution:<br /></b><br />Using G(x) : x is a giant.<br /><br />M(x) : x is a man.<br /><br />The given statement can be symbolized as (x)(M(x)<span style="font-family:Symbol;">®</span> G(x)).<br /><br />If we restrict the variable x to the universe, which is the class of men, then the statement is (x)G(x).<br /><br /><br /><b><br />Example 2:<br /></b><br />Consider the statement "Given any positive integer, there is a greater positive integer". Symbolize this statement with and without using the set of positive integer as the universe of discourse.<br /><b><br />Solution:<br /></b><br />Let the variables x and y be restricted to the set of positive integers.<br /><br />For all x, there exists a y such that y is greater than x. If G(x, y) is "x is greater than y", then the given statement is (x)(<span style="font-family:Symbol;">$</span>y)G(y, x).<br /><br />If we do not impose the restriction on the universe of discourse and if we write P(x) for "x is a positive integer" then we can symbolize the given statement as (x)(P(x)<span style="font-family:Symbol;">®</span>(<span style="font-family:Symbol;">$</span>y)(P(y)<span style="font-family:Symbol;">Ù</span>G(y, x))).<br /><br />If the correct connectives are not used, then the meaning changes.<br /><br /><b><br />Examples:<br /><br /></b>1. All cats are animals.<br /><br />This is true for any universe of discourse. In particular let the universe of discourse E be {Cuddle,Ginger,0,1}, where the first two elements are the names of cats.<br /><br />The statement (1) is true over E.<br /><br />Now consider the statements<br /><br />(x)(C(x)<span style="font-family:Symbol;">®</span> A(x)) and (x)(C(x)<span style="font-family:Symbol;"> Ù </span>A(x))<br /><br />where, C(x) : x is a cat.<br /><br />A(x) : x is an animal.<br /><br />In (x)(C(x)<span style="font-family:Symbol;">®</span>A(x)), if x is replaced by any of the elements of E, then we get a true statement; hence (x)(C(x)<span style="font-family:Symbol;">®</span>A(x)) is true over E.<br /><br />When x is replaced by 0 or 1 (x)(C(x)<span style="font-family:Symbol;">Ù</span>A(x)) becomes false over E because C(x)<span style="font-family:Symbol;">Ù</span>A(x) assumes false. Therefore, the statement (1) can not be symbolized as (x)(C(x)<span style="font-family:Symbol;">Ù</span>A(x))<br /><br />Now consider the statement<br /><br />2. Some cats are black.<br /><br />Let E be as above and assume both Cuddle and Ginger are white cats, and let B(x) : x is black.<br /><br />In this case there is no black cat in the universe E, and (2) is false.<br /><br />The statement (<span style="font-family:Symbol;">$</span> x)(C(x)<span style="font-family:Symbol;">Ù</span> B(x)) is also false over E because there is no black cat in E.<br /><br /><br />On the other hand, (<span style="font-family:Symbol;">$</span> x)(C(x)<span style="font-family:Symbol;">®</span> B(x)) is true because C(x)<span style="font-family:Symbol;">®</span> B(x) is true when x is replaced by 0 or 1. Therefore the conditional is not the correct connective to use in this case.Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1tag:blogger.com,1999:blog-7399621945608861143.post-58420279448351314622008-08-12T04:21:00.003-07:002008-12-23T01:55:02.555-08:00Automatic Theorem Proving<b><a name="Automatic Theorem Proving"></a></b><br /> <br />The system consists of 10 rules, an axiom schema, and rules of well formed sequents and formulas.<br /> <br />1.<b> Variables :</b><br /> <br />The capital letters A, B, C, …, P, Q, R, … are used as statement variables. They are also used as statement formulas.<br /> <br />2. <b>Connectives:</b><br /> <br />The connectives <span style="font-family:Symbol;">ù</span> , <span style="font-family:Symbol;">Ù</span> , <span style="font-family:Symbol;">Ú</span> , <span style="font-family:Symbol;">®</span> , and <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> appear in the formulas with the order of precedence as given; <span style="font-family:Symbol;">ù</span> has the highest precedence, followed by <span style="font-family:Symbol;">Ù</span> , and so on.<br /> <br />3. <b>String of Formulas :<br /> </b> <br />A string of formulas is defined as follows:<br /> <ol type="a"><ol type="a"><li> <p style="line-height: 150%;">Any formula is a string of formulas. </p></li><li> <p style="line-height: 150%;">If <span style="font-family:Symbol;">a</span> and <span style="font-family:Symbol;">b</span> are strings of formulas, then <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> and <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">a</span> are strings of formulas. </p></li><li> <p style="line-height: 150%;">Only those strings which are obtained by steps (a) and (b) are strings of formulas, with the exceptions of the empty string which is also a string of formulas. </p></li></ol></ol> <br />Example, the strings A, B, C; B,C,A; A,C,B, etc are the same.<br /> <br />4. <b>Sequents :</b><br /> <br />If <span style="font-family:Symbol;">a</span> and <span style="font-family:Symbol;">b</span> are string of formulas, then <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image118.gif" width="42" align="absmiddle" height="32" /> is called a sequent in which <span style="font-family:Symbol;">a</span> is denoted the antecedent and <span style="font-family:Symbol;">b</span> the consequent of the sequent.<br /> <br />Sequent <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image118.gif" width="42" align="absmiddle" height="32" /> is true iff either atleast one of the formulas of the antecedent is false or atleast one of the formulas of the consequent is true. Thus<br /> <br />A, B, C,<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image119.gif" width="46" height="29" />, E, F is true iff A<span style="font-family:Symbol;">Ù</span> B<span style="font-family:Symbol;">Ù</span> C<span style="font-family:Symbol;">®</span> D<span style="font-family:Symbol;">Ú</span> E<span style="font-family:Symbol;">Ú</span> F is true.<br /> <br />The symbol <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image120.gif" width="20" align="absmiddle" height="29" /> is a generalization of the connective <span style="font-family:Symbol;">®</span> to strings of formulas.<br /> <br />A<span style="font-family:Symbol;">Þ</span> B means "A implies B" or "A<span style="font-family:Symbol;">®</span> B is a tautology" while <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image121.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> means that <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image120.gif" width="20" align="absmiddle" height="29" /> <span style="font-family:Symbol;">b</span> is true.<br /> <br />For example P, Q, R<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image122.gif" width="20" align="absmiddle" height="29" />P, N.<br /> <br />The empty antecedent is interpreted as the logical constant "true" or T, and empty consequent is interpreted as the logical constant "false" or F.<br /> <br />5. <b>Axiom Schema :<br /> </b> <br />If <span style="font-family:Symbol;">a</span> and <span style="font-family:Symbol;">b</span> are strings of formulas such that every formula in both <span style="font-family:Symbol;">a</span> and <span style="font-family:Symbol;">b</span> is a variable only, then the sequent <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image123.gif" width="42" align="absmiddle" height="32" /> is an axiom iff <span style="font-family:Symbol;">a</span> and <span style="font-family:Symbol;">b</span> have atleast one variable in common. As an example, A,B,C<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image124.gif" width="20" align="absmiddle" height="29" />P,B, R, where A, B, C, P and R are variables, is an axiom<br /> <br />If <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image120.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> is an axiom, then <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image125.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> .<br /> <br />6. <b>Theorem :<br /> </b><ol type="a"><b> </b><ol type="a"><b> </b><li> <p style="line-height: 150%;">Every axiom is a theorem. </p></li><li> <p style="line-height: 150%;">If a sequent <span style="font-family:Symbol;">a</span> is a theorem and a sequent <span style="font-family:Symbol;">b</span> results from <span style="font-family:Symbol;">a</span> through the use of one of the 10 rules of the system, which are given below, then <span style="font-family:Symbol;">b</span> is a theorem. </p></li><li> <p style="line-height: 150%;">Sequents obtained by (a) and (b) are the only theorem. </p></li></ol></ol> <br />7. <b>Rules :</b><br /> <br />The following rules are used to combine formulas within strings by introducing connectives. Corresponding to each of the connectives there are two rules, one for the introduction of the connective in the antecedent and the other for its introduction in the consequent. In the description of these rules <span style="font-family:Symbol;">a</span>,<span style="font-family:Symbol;">b</span>,<span style="font-family:Symbol;">g</span>,… are strings of formulas while X and Y are formulas to which the connectives are applied.<br /> <br /><br /> <nobr> <b> <br />Antecedent Rules<br /> <br /> </b>Rule <span style="font-family:Symbol;">ù</span> <span style="font-family:Symbol;">Þ</span> : If <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />X, <span style="font-family:Symbol;">g</span> , then <span style="font-family:Symbol;">a</span> ,<span style="font-family:Symbol;">ù</span> X, <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> .<br /> Rule <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">Þ</span> : If X, Y, <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> , then <span style="font-family:Symbol;">a</span> , X<span style="font-family:Symbol;">Ù</span> Y, <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> .<br /> Rule <span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">Þ</span> : If X, <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> and Y, <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> , then <span style="font-family:Symbol;">a</span> , X <span style="font-family:Symbol;">Ú</span> Y, <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> .<br /> Rule <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> : If Y, <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g</span> and <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />X, <span style="font-family:Symbol;">g </span>then <span style="font-family:Symbol;">a</span> , X<span style="font-family:Symbol;">®</span> Y, <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g </span>.<br /> Rule <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> <span style="font-family:Symbol;">Þ</span> : If X, Y, <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g </span>and <span style="font-family:Symbol;">a</span> , <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />X, Y, <span style="font-family:Symbol;">g </span>, then <span style="font-family:Symbol;">a</span> , X <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Y, <span style="font-family:Symbol;">b</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">g </span>.<br /> <b><br /> Consequent Rules<br /> <br /> </b>Rule <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù </span>: If X, <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>, then <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">ù</span>X, <span style="font-family:Symbol;">g .<br /> </span>Rule <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">Ù</span> : If <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />X, <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>and <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />Y, <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>, then <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> , X<span style="font-family:Symbol;">Ù</span> Y, <span style="font-family:Symbol;">g </span>.<br /> Rule <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">Ú</span> : If <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />X, Y, <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>then <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> , X <span style="font-family:Symbol;">Ú</span> Y, <span style="font-family:Symbol;">g </span>.<br /> Rule <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> : If X, <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />Y, <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>then <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> , X<span style="font-family:Symbol;">®</span> Y, <span style="font-family:Symbol;">g .<br /> </span>Rule <span style="font-family:Symbol;">Þ</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> : If X, <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />Y, <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>and Y, <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />X, <span style="font-family:Symbol;">b</span> , <span style="font-family:Symbol;">g </span>, then <span style="font-family:Symbol;">a</span> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">b</span> , X<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Y, <span style="font-family:Symbol;">g </span>.<br /> <br /> In order to show that C follows from H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub> we establish that<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image127.gif" width="20" align="absmiddle" height="29" /> H<sub>1</sub><span style="font-family:Symbol;">®</span> <sub> </sub>(H<sub>2</sub><span style="font-family:Symbol;">®</span> <sub> </sub>(H<sub>3</sub>.......(H<sub>m</sub><span style="font-family:Symbol;">®</span> <sub> </sub>C)....)) ………… (1)<br /> <br />is a theorem we must show that<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image128.gif" width="20" align="absmiddle" height="29" /> H<sub>1</sub><span style="font-family:Symbol;">®</span> <sub> </sub>(H<sub>2</sub><span style="font-family:Symbol;">®</span> <sub> </sub>(H<sub>3</sub>....(H<sub>m</sub><span style="font-family:Symbol;">®</span> <sub> </sub>C).....)) ………… (2)<br /> <br />our procedure involves showing (1) to be a theorem.<br /> <br />We first assume (2) and then show that this assumption is or is not justified. This task is accomplished by working backward from (2), using the rules, and showing that (2) holds if some simpler sequent is a theorem. We continue working backward until we arrive at the simplest possible sequents i.e., those which do not have any connectives.<br /> <br />If these sequents are axioms, then we have justified our assumption of (2). If atleast one of the simplest sequent is not an axiom, then the assumption of (2) is not justified and C does not follows from H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub>. In the case C follows from H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub> the derivation of (2) is easily constructed by simply working through the same steps, starting from the axioms obtained.<br /> <b> <br />Example 1:<br /> </b> <br />Show that P<span style="font-family:Symbol;">Ú</span>Q follows from P.<br /> <b> <br />Solution:<br /> </b> <br />We need to show that<br /><br /> (1) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image129.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ú</span> Q)<br /> (1) if (2) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">Ú</span> Q (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> )<br /> (2) if (3) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P, Q (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">Ú</span> )<br /><br />We first eliminate the connective <span style="font-family:Symbol;">®</span> in (1). Using the rule <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> we have "if P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">Ú</span> Q then <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ú</span> Q)". Here we have named P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">Ú</span> Q by (2). Each line of derivation thus introduces the name as well as gives a rule "(1) if (2)" means "if (2) then (1)". The chain of arguments is then given by (1) holds if (2), and (2) holds if (3). Finally (3) is a theorem, because it is an axiom. (3) is an axiom that leads to <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ú</span> Q) as shown.<br /><br />(a) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P, Q Axiom<br />(b) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">Ú</span> Q Rule (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">Ú</span> ), (a)<br /> (c) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P<span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ú</span> Q) Rule (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> ), (b).<br /><br /><b> <br />Example 2:<br /> </b> <br />Show that <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />(<span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> Q)) <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>P.<br /> <b> <br />Solution:<br /> <br /> </b> (1) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />(<span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> Q))<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>P <br />(1) if (2) <span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> Q) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">ù</span> P (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> )<br /> (2) if (3) <span style="font-family:Symbol;">ù</span>Q, P<span style="font-family:Symbol;">®</span> Q<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">ù</span>P (<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">Þ</span> )<br />(3) if (4) P<span style="font-family:Symbol;">®</span> Q<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">ù</span>P, Q (<span style="font-family:Symbol;">ù Þ </span>)<br />(4) if (5) Q<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /><span style="font-family:Symbol;">ù</span>P, Q and (6) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P, <span style="font-family:Symbol;">ù</span>P, Q (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br />(5) if (7) P, Q <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />Q (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span> )<br />(6) if (8) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P, Q (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span> )<br /> <br />Now (7) and (8) are axioms, hence the theorem (1) follows.<br /> <br /><br /> <b> Example 3:<br /> </b> <br />Does P follows from P <span style="font-family:Symbol;">Ú</span> Q?<br /> <b> <br />Solution:<br /> </b> <br />We investigate whether <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />(P<span style="font-family:Symbol;">Ú</span> Q) <span style="font-family:Symbol;">®</span> P is a theorem.<br />Assume (1) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />(P<span style="font-family:Symbol;">Ú</span> Q)<span style="font-family:Symbol;">®</span> P<br /> (1) if (2) (P <span style="font-family:Symbol;">Ú</span> Q) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> )<br /> (2) if (3) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P and (4) Q <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P (<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">Þ</span> ) <br />Here (3) is an axiom, but (4) is not. Hence P does not follow from P <span style="font-family:Symbol;">Ú</span> Q.<br /> <b> <br />Example 4:<br /> </b> <br />Show that S<span style="font-family:Symbol;">Ú</span> R is tautologically implied by (P<span style="font-family:Symbol;">Ú</span> Q) <span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> R)<span style="font-family:Symbol;">Ù</span> (Q<span style="font-family:Symbol;">®</span> S).<br /> <b> <br />Solution:<br /> </b> <br />To show (1) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />((P<span style="font-family:Symbol;">Ú</span> Q)<span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> R)) <span style="font-family:Symbol;">Ù</span> (Q<span style="font-family:Symbol;">®</span> S))<span style="font-family:Symbol;">®</span> (S<span style="font-family:Symbol;">Ú</span> R).<br /> (1) if (2) (P<span style="font-family:Symbol;">Ú</span> Q)<span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> R)<span style="font-family:Symbol;">Ù</span> (Q<span style="font-family:Symbol;">®</span> S) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />(S<span style="font-family:Symbol;">Ú</span> R) (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">®</span> )<br />(2) if (3) (P<span style="font-family:Symbol;">Ú</span> Q)<span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> R)<span style="font-family:Symbol;">Ù</span> (Q<span style="font-family:Symbol;">®</span> S) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /> S, R (<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">Ú</span> ) <br />(3) if (4) (P<span style="font-family:Symbol;">Ú</span> Q), (P<span style="font-family:Symbol;">®</span> R), (Q<span style="font-family:Symbol;">®</span> S) <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S, R (<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">Þ</span> twice)<br />(4) if (5) P, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S,R and (6) Q, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S,R (<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">Þ</span> )<br />(5) if (7) P, R, Q<span style="font-family:Symbol;">®</span> S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S,R and (8) P, Q<span style="font-family:Symbol;">®</span> S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P,S,R (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br />(6) if (9) P, R, S, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S, R and (10) P, R <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S,R,Q (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br />(7) if (11) P, S <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P, S, R and (12) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />P, S, R, Q (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br />(8) if (13) Q, R, Q<span style="font-family:Symbol;">®</span> S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S, R and (14) Q, Q<span style="font-family:Symbol;">®</span> S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S, R, P (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br />(9) if (15) Q, R, S<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S, R and (16) Q, R <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /> S, R, Q (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br />(10) if (17) Q, S <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" />S, R, P and (18) Q <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Image126.gif" width="20" align="absmiddle" height="29" /> S, R, P, Q (<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">Þ</span> )<br /><br />Now (9) to (12) and (15) to (18) are all axioms. Therefore, the result follows.<br /></nobr> <br /><br /> <p style="line-height: 150%;" align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Section1.4.htm#1.4%20Automatic%20Theorem%20Proving">Back to top</a></span><br /> </p><p style="line-height: 150%;" align="left"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.4/Image/Sectio1.gif" width="768" border="0" height="10" /><br /> <b> <br /><br /> <br /><span style="font-size:130%;"><a name="Exercise :">Exercise:</a></span><br /> </b> <nobr> <br /> 1. Show the validity of the following arguments for which the premises are given on the left and the conclusion on the right. <blockquote> <ol type="a"><li> <span style="font-family:Symbol;">ù</span>(P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>Q), <span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">Ú</span> R, <span style="font-family:Symbol;">ù</span>R <span style="font-family:Symbol;">ù</span>P</li><li>(A<span style="font-family:Symbol;">®</span> B)<span style="font-family:Symbol;">Ù</span> (A<span style="font-family:Symbol;">®</span> C), <span style="font-family:Symbol;">ù</span>(B<span style="font-family:Symbol;">Ù</span> C), D<span style="font-family:Symbol;">Ú</span> A D</li><li> <span style="font-family:Symbol;">ù</span>J<span style="font-family:Symbol;">®</span> (M<span style="font-family:Symbol;">Ú</span> N), (H<span style="font-family:Symbol;">Ú</span> G)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>J, H<span style="font-family:Symbol;">Ú</span> G M<span style="font-family:Symbol;">Ú</span> N</li><li> P<span style="font-family:Symbol;">®</span> Q, (<span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">Ú</span> R) <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>R, <span style="font-family:Symbol;">ù</span>(<span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ù</span> S) <span style="font-family:Symbol;">ù</span>S</li><li>(P<span style="font-family:Symbol;">Ù</span> Q)<span style="font-family:Symbol;">®</span> R, <span style="font-family:Symbol;">ù</span>R<span style="font-family:Symbol;">Ú</span> S, <span style="font-family:Symbol;">ù</span>S <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span>Q</li><li>P<span style="font-family:Symbol;">®</span> Q, Q<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R, R, P<span style="font-family:Symbol;">Ú</span> (J<span style="font-family:Symbol;">Ù</span> S) J<span style="font-family:Symbol;">Ù</span> S</li><li>B<span style="font-family:Symbol;">Ù</span> C, (B<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> C)<span style="font-family:Symbol;">®</span> (H<span style="font-family:Symbol;">Ú</span> G) G<span style="font-family:Symbol;">Ú</span> H</li><li>(P<span style="font-family:Symbol;">®</span> Q)<span style="font-family:Symbol;">®</span> R, P<span style="font-family:Symbol;">Ù</span> S, Q<span style="font-family:Symbol;">Ù</span> T R<br /> </li></ol> </blockquote><br /> 2. Derive the following, using rule CP if necessary <ol type="a"><ol type="a"><li> <p style="line-height: 150%;"> <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ú</span> Q, <span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">Ú</span> R, R<span style="font-family:Symbol;">®</span> S <span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">®</span> S. </p></li><li> <p style="line-height: 150%;">P, P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> (R<span style="font-family:Symbol;">Ù</span> S)) <span style="font-family:Symbol;">Þ</span> Q<span style="font-family:Symbol;">®</span> S. </p></li><li> <p style="line-height: 150%;">P<span style="font-family:Symbol;">®</span> Q <span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ù</span> Q). </p></li><li> <p style="line-height: 150%;">(P<span style="font-family:Symbol;">Ú</span> Q)<span style="font-family:Symbol;">®</span> R <span style="font-family:Symbol;">Þ</span> (P<span style="font-family:Symbol;">Ù</span> Q)<span style="font-family:Symbol;">®</span> R. </p></li><li> <p style="line-height: 150%;">P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> R), Q<span style="font-family:Symbol;">®</span> (R<span style="font-family:Symbol;">®</span> S) <span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> S). </p></li></ol></ol> <br /> 3. Show that the following sets of premises are inconsistent. <ol type="a"><ol type="a"><li> <p style="line-height: 150%;">P<span style="font-family:Symbol;">®</span> Q, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R, P. </p></li><li> <p style="line-height: 150%;">A<span style="font-family:Symbol;">®</span> (B<span style="font-family:Symbol;">®</span> C), D<span style="font-family:Symbol;">®</span> (B<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>C), A<span style="font-family:Symbol;">Ù</span> D. </p></li></ol></ol> <br /> Hence show that P<span style="font-family:Symbol;">®</span> Q, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R, P<span style="font-family:Symbol;">Þ</span> M, and A<span style="font-family:Symbol;">®</span> (B<span style="font-family:Symbol;">®</span> C), D<span style="font-family:Symbol;">®</span> (B<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>C), A<span style="font-family:Symbol;">Ù</span> D<span style="font-family:Symbol;">Þ</span> P.<br /> <br /> 4. Show the following (use indirect method if needed)<br /> <blockquote> <ol type="a"><li> (R<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>Q), R<span style="font-family:Symbol;">Ú</span> S, S<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>Q, P<span style="font-family:Symbol;">®</span> Q<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span>P.</li><li> S<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>Q, S<span style="font-family:Symbol;">Ú</span> R, <span style="font-family:Symbol;">ù</span>R, <span style="font-family:Symbol;">ù</span>R<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Q<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span>P.</li><li> <span style="font-family:Symbol;">ù</span>(P<span style="font-family:Symbol;">®</span> Q)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>(R<span style="font-family:Symbol;">Ú</span> S), ((Q<span style="font-family:Symbol;">®</span> P)<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span>R), R<span style="font-family:Symbol;">Þ</span> P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Q.</li></ol> </blockquote> <br /> 5. Show the following<br /> <ol type="a"><ol type="a"><li> P<span style="font-family:Symbol;">Þ</span> (<span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">®</span> Q).</li><li> P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ù</span> Q<span style="font-family:Symbol;">Þ</span> R.</li><li> R<span style="font-family:Symbol;">Þ</span> (P<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ú</span> Q)</li><li> <span style="font-family:Symbol;">ù </span>(P<span style="font-family:Symbol;">Ù</span> Q)<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span>Q</li></ol></ol> </nobr> </p>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com0tag:blogger.com,1999:blog-7399621945608861143.post-80915546880066991882008-08-12T04:21:00.001-07:002008-12-23T01:55:02.556-08:00Theory of Inference<b><a name="Theory of Inference"></a></b><br /><i><br />Theory, which is associated with the inferring of conclusion from the given set of premises using accepted rules of reasoning, is called the <b>theory of inference</b>.</i><br /><i><br />The process of derivation of a conclusion from the given set of premises using the rules of inference is known as <b>formal proof or deduction</b>.<br /></i><br />In a formal proof, every rule of inference that is used at any stage in the derivation is acknowledged. The rules of implications and equivalences are used as the accepted rules in inference theory.<br /><br /><br /><b><br /><a name="1.3.1 Validity Using Truth Tables">1.3.1 Validity Using Truth Tables</a><br /></b><br />Let A and B two statement formulas we say that "B logically follows from A" or "B is a valid conclusion (consequence) of the premise A" iff A<span style="font-family:Symbol;">®B</span> is a tautology. That is, <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/Image109.gif" width="50" align="absbottom" height="18" />. From a set of premises {H<sub>1</sub>, H<sub>2</sub>, . . ., H<sub>m</sub>} a conclusion C follows logically iff <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/Image111.gif" width="162" align="absmiddle" height="22" />.<br /><br /><br /><span style="font-size:85%;"> </span><p align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Section1.3.htm#1.3%20Theory%20of%20Inference">Back to top</a><br /></span></p><p><span style="font-size:85%;"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Image/Sectio11.gif" width="772" border="0" height="10" /> </span><b><br /><br /><br /><a name="1.3.2 Testing the Validity of a Conclusion">1.3.2 Testing the Validity of a Conclusion</a><br /></b><br />Let P<sub>1</sub>, P<sub>2</sub>, … , P<sub>n</sub> be all the atomic variables appearing in the premises H<sub>1</sub>,H<sub>2</sub>, … , H<sub>m</sub> and the conclusion C.<br /><br />In order to test the validity of a conclusion C from the premises H<sub>1</sub>, H<sub>2</sub>, … , H<sub>m</sub> using truth table, we assign all possible combinations of truth values to all variables P<sub>1</sub>, P<sub>2</sub>, … , P<sub>n</sub> that occur in H<sub>1</sub>, H<sub>2</sub>, … , H<sub>m</sub> and C. Construct truth table of the premises and conclusion. Look for the rows in which all the H<sub>i</sub>’s are true (T). If for every such row C is true, then C is a valid conclusion from the premises and hence H<sub>1</sub><span style="font-family:Symbol;">Ù</span> H<sub>2</sub><span style="font-family:Symbol;">Ù</span> …<span style="font-family:Symbol;">Ù</span> H<sub>n</sub> <span style="font-family:Symbol;">Þ</span> C or look for rows in which C has a truth value false (F). If, in every such row, atleast one of the values of H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub> is F, then C is a valid conclusion from the premises.<br /><b> <br /><br /> <br />Example 1:<br /> </b> <br />Determine whether the conclusion C follows logically from the premises H<sub>1</sub> and H<sub>2</sub>.<br /><br />(a) <b> </b> <b>H<sub>1</sub></b> : P<span style="font-family:Symbol;">®</span> Q <b> </b> <b>H<sub>2</sub></b> : P <b> </b> <b>C</b> : Q<br /><table width="330" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="62"><b> </b><p align="center"><b>P</b></p></td> <td valign="top" width="54"><b> </b><p align="center"><b>Q</b></p></td> <td valign="top" width="57"><b> </b><p align="center"><b>H<sub>1</sub></b></p></td> <td valign="top" width="33"><b> </b><p align="center"><b>H<sub>2</sub></b></p></td> <td valign="top" width="30"><b> </b><p align="center"><b>C</b></p></td> </tr> <tr> <td valign="top" width="62"> <p align="center">T</p></td> <td valign="top" width="54"> <p align="center">T</p></td> <td valign="top" width="57"> <p align="center">T</p></td> <td valign="top" width="33"> <p align="center">T</p></td> <td valign="top" width="30"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="62"> <p align="center">T</p></td> <td valign="top" width="54"> <p align="center">F</p></td> <td valign="top" width="57"> <p align="center">F</p></td> <td valign="top" width="33"> <p align="center">T</p></td> <td valign="top" width="30"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="62"> <p align="center">F</p></td> <td valign="top" width="54"> <p align="center">T</p></td> <td valign="top" width="57"> <p align="center">F</p></td> <td valign="top" width="33"> <p align="center">F</p></td> <td valign="top" width="30"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="62"> <p align="center">F</p></td> <td valign="top" width="54"> <p align="center">F</p></td> <td valign="top" width="57"> <p align="center">T</p></td> <td valign="top" width="33"> <p align="center">F</p></td> <td valign="top" width="30"> <p align="center">F</p></td> </tr> </tbody></table><br /><br /> <br />If H<sub>1</sub> and H<sub>2</sub> have truth values T, then check C. If C has truth value T, then<br /> <br />H<sub>1</sub> <span style="font-family:Symbol;">Ù</span> H<sub>2</sub> <span style="font-family:Symbol;">Þ</span> C. The first row (in H<sub>1</sub> and H<sub>2</sub>) in which both premises have the value T. The conclusion C has the value T in that row. Therefore H<sub>1</sub><span style="font-family:Symbol;">Ù</span> H<sub>2</sub> <span style="font-family:Symbol;">Þ</span> C. It is a valid conclusion.<br /> <br /><br /><br />(b) <b> </b> <b>H<sub>1</sub></b> : P<span style="font-family:Symbol;">®</span> Q <b> </b> <b>H<sub>2</sub></b> : <span style="font-family:Symbol;">ù</span> P <b> </b> <b>C</b> : Q<br /><table width="348" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="18%"><b> </b><p align="center"><b>P</b></p></td> <td valign="top" width="22%"><b> </b><p align="center"><b>Q</b></p></td> <td valign="top" width="22%"><b> </b><p align="center"><b>H<sub>1</sub></b></p></td> <td valign="top" width="19%"><b> </b><p align="center"><b>H<sub>2</sub></b></p></td> <td valign="top" width="19%"><b> </b><p align="center"><b>C</b></p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> </tbody></table> <br />Check the third and fourth row. The conclusion Q is true only in the third row, but not in the fourth.<br /> <br />Hence the conclusion is not valid.<br /> <br /><br /><br />(c) <b> </b> <b>H<sub>1</sub></b> : P<span style="font-family:Symbol;">®</span> Q <b> </b> <b>H<sub>2</sub></b> : <span style="font-family:Symbol;">ù</span> (P<span style="font-family:Symbol;">Ù</span> Q) <b> </b> <b>C</b> : <span style="font-family:Symbol;">ù</span> P<br /><table width="348" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="18%"><b> </b><p align="center"><b>P</b></p></td> <td valign="top" width="22%"><b> </b><p align="center"><b>Q</b></p></td> <td valign="top" width="22%"><b> </b><p align="center"><b>H<sub>1</sub></b></p></td> <td valign="top" width="19%"><b> </b><p align="center"><b>H<sub>2</sub></b></p></td> <td valign="top" width="19%"><b> </b><p align="center"><b>C</b></p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> </tr> </tbody></table> <br /><br /> <br />Check third and fourth row, the conclusion C has true in both rows. Therefore it is a valid conclusion.<br /> <br /><br /><br />(d) <b> </b> <b>H<sub>1</sub></b> : <span style="font-family:Symbol;">ù</span> P <b> </b> <b>H<sub>2</sub></b> : P<span style="font-family:Symbol;">Ú</span> Q <b> </b> <b>C</b> : P<span style="font-family:Symbol;">Ù</span> Q<br /><table width="348" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="18%"><b> </b><p align="center"><b>P</b></p></td> <td valign="top" width="22%"><b> </b><p align="center"><b>Q</b></p></td> <td valign="top" width="22%"><b> </b><p align="center"><b>H<sub>1</sub></b></p></td> <td valign="top" width="19%"><b> </b><p align="center"><b>H<sub>2</sub></b></p></td> <td valign="top" width="19%"><b> </b><p align="center"><b>C</b></p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="18%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">F</p></td> <td valign="top" width="22%"> <p align="center">T</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> <td valign="top" width="19%"> <p align="center">F</p></td> </tr> </tbody></table> <br /><br /> <br />Check the third row. The conclusion C has value F therefore it is not a valid conclusion.<br /> <br /><br /><br />(e) <b> </b> <b>H<sub>1</sub></b> : P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> R) <b> </b> <b>H<sub>2</sub></b> : R <b> </b> <b>C</b> : P<br /><table width="460" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="15%"><b> </b><p align="center"><b>P</b></p></td> <td valign="top" width="17%"><b> </b><p align="center"><b>Q</b></p></td> <td valign="top" width="14%"><b> </b><p align="center"><b>R</b></p></td> <td valign="top" width="14%"><b> </b><p align="center"><b>Q<span style="font-family:Symbol;">®</span> R</b></p></td> <td valign="top" width="12%"><b> </b><p align="center"><b>H<sub>1</sub></b></p></td> <td valign="top" width="14%"><b> </b><p align="center"><b>H<sub>2</sub></b></p></td> <td valign="top" width="12%"><b> </b><p align="center"><b>C</b></p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">T</p></td> <td valign="top" width="17%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">T</p></td> <td valign="top" width="17%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">T</p></td> <td valign="top" width="17%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">T</p></td> <td valign="top" width="17%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">F</p></td> <td valign="top" width="17%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">F</p></td> <td valign="top" width="17%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">F</p></td> <td valign="top" width="17%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="15%"> <p align="center">F</p></td> <td valign="top" width="17%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="14%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="14%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> </tbody></table> <br /><br /> <br />Check first, third, fifth and seventh rows. But C has truth value T only in first, third but not in fifth and seventh. Therefore, it is not a valid conclusion.<br /> <br /><br /><br />(f) <b> </b> <b>H<sub>1</sub></b> : P<span style="font-family:Symbol;">Ú</span> Q <b> </b> <b>H<sub>2</sub></b> : P<span style="font-family:Symbol;">®</span> R <b> </b> <b>H<sub>3</sub></b> : Q<span style="font-family:Symbol;">®</span> R <b> </b> <b>C</b> : R<br /><table width="490" border="1" cellpadding="7" cellspacing="1"> <tbody><tr> <td valign="top" width="16%"><b> </b><p align="center"><b>P</b></p></td> <td valign="top" width="16%"><b> </b><p align="center"><b>Q</b></p></td> <td valign="top" width="16%"><b> </b><p align="center"><b>R</b></p></td> <td valign="top" width="12%"><b> </b><p align="center"><b>H<sub>1</sub></b></p></td> <td valign="top" width="13%"><b> </b><p align="center"><b>H<sub>2</sub></b></p></td> <td valign="top" width="13%"><b> </b><p align="center"><b>H<sub>3</sub></b></p></td> <td valign="top" width="12%"><b> </b><p align="center"><b>C</b></p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">F</p></td> <td valign="top" width="13%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">F</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">T</p></td> </tr> <tr> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="16%"> <p align="center">F</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="13%"> <p align="center">T</p></td> <td valign="top" width="12%"> <p align="center">F</p></td> </tr> </tbody></table> <br /><br /><br />Check first, third and fifth rows. In all the above said rows H<sub>1</sub>, H<sub>2</sub>, H<sub>3</sub> and C have truth value T. Therefore, it is a valid conclusion.<br /><br /><br /><span style="font-size:85%;"> </span></p><p align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Section1.3.htm#1.3%20Theory%20of%20Inference">Back to top</a><br /></span> </p><p><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Image/Sectio11.gif" width="766" border="0" height="10" /><br /><br /><b> <a name="1.3.3 Rules of Inference"> 1.3.3 Rules of Inference<br /> </a> </b> <br />Consider<br /> <b> <br />Rule P </b>: A premise may be introduced at any point in the derivation.<br /> <b> <br />Rule T </b>: A formula S may be introduced in a derivation if S is tautologically implied by one or more of the preceding formulas in the derivation.<br /> <br /><br /> <b> </b></p><p align="center"><b>Table 1.3.3 : Implications<br /> </b> <br /> </p><p align="left"><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/1.3.1.gif" width="233" border="0" height="50" /><br /> <br /> <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/1.3.2.gif" width="232" border="0" height="51" /><br /> <br /> <nobr> I<sub>5</sub> <span style="font-family:Symbol;"> ù</span> P<span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">®</span> Q<br /> <br /> I<sub>6</sub> Q <span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">®</span> Q<br /> <br /> I<sub>7</sub> <span style="font-family:Symbol;"> ù </span>(P<span style="font-family:Symbol;">®</span> Q) <span style="font-family:Symbol;">Þ</span> P<br /> <br /> I<sub>8</sub> <span style="font-family:Symbol;"> ù</span> (P<span style="font-family:Symbol;">®</span> Q) <span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span> Q<br /> <br /> I<sub>9</sub> P,Q <span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">Ù</span> Q<br /> <br /> I<sub>10</sub> <span style="font-family:Symbol;">ù</span> P,P <span style="font-family:Symbol;">Ú</span> Q<span style="font-family:Symbol;">Þ</span> Q<br /> <br /> I<sub>11</sub> P, P<span style="font-family:Symbol;">®</span> Q<span style="font-family:Symbol;">Þ</span> Q<br /> <br /> I<sub>12</sub> <span style="font-family:Symbol;"> ù</span> Q, P<span style="font-family:Symbol;">®</span> Q<span style="font-family:Symbol;">Þ</span> <span style="font-family:Symbol;">ù</span> P<br /> <br /> I<sub>13</sub> P<span style="font-family:Symbol;">®</span> Q, Q<span style="font-family:Symbol;">®</span> R <span style="font-family:Symbol;">Þ</span> P<span style="font-family:Symbol;">®</span> R<br /> <br /> I<sub>14</sub> P<span style="font-family:Symbol;">Ú</span> Q, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> R <span style="font-family:Symbol;">Þ</span> R<br /> </nobr><br /> </p><p align="center"> <b> Table 1.3.4 : Equivalences<br /> </b> </p><p>E<sub>1</sub> <span style="font-family:Symbol;"> ù</span> <span style="font-family:Symbol;">ù</span> P <span style="font-family:Symbol;">Û</span> P<br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/1.3.3.gif" width="286" border="0" height="50" /><br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/1.3.4.gif" width="393" border="0" height="50" /><br /> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/1.3.5.gif" width="468" border="0" height="50" /><br /><br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Image/1.3.6.gif" width="310" border="0" height="56" /><br /> <br /><nobr>E<sub>10</sub> P <span style="font-family:Symbol;">Ú</span> P <span style="font-family:Symbol;">Û</span> P<br /> <br />E<sub>11</sub> P <span style="font-family:Symbol;">Ù</span> P <span style="font-family:Symbol;">Û</span> P<br /> <br />E<sub>12</sub> R <span style="font-family:Symbol;">Ú</span> (P <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span> P) <span style="font-family:Symbol;">Û</span> R<br /> <br />E<sub>13</sub> R <span style="font-family:Symbol;">Ù</span> (P <span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span> P) <span style="font-family:Symbol;">Û</span> R<br /> <br />E<sub>14</sub> R <span style="font-family:Symbol;">Ú</span> (P <span style="font-family:Symbol;">Ú</span> <span style="font-family:Symbol;">ù</span>P) <span style="font-family:Symbol;">Û</span> T<br /> <br />E<sub>15</sub> R <span style="font-family:Symbol;">Ù</span> (P <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>P) <span style="font-family:Symbol;">Û</span> F.<br /> <br />E<sub>16</sub> P <span style="font-family:Symbol;">®</span> Q <span style="font-family:Symbol;">Û</span> <span style="font-family:Symbol;">ù</span> P <span style="font-family:Symbol;">Ú</span> Q<br /><br />E<sub>17</sub> <span style="font-family:Symbol;"> ù</span> (P <span style="font-family:Symbol;">®</span> Q) <span style="font-family:Symbol;">Û</span> P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span> Q<br /> <br />E<sub>18</sub> P <span style="font-family:Symbol;">®</span> Q <span style="font-family:Symbol;">Û</span> <span style="font-family:Symbol;">ù</span> Q <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span> P<br /> <br />E<sub>19</sub> P <span style="font-family:Symbol;">®</span> (Q <span style="font-family:Symbol;">®</span> R) <span style="font-family:Symbol;">Û</span> (P<span style="font-family:Symbol;">Ù</span> Q)<span style="font-family:Symbol;">®</span> R<br /><br />E<sub>20</sub> <span style="font-family:Symbol;"> ù</span> (P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Q) <span style="font-family:Symbol;">Û</span> P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> <span style="font-family:Symbol;">ù</span> Q<br /><br />E<sub>21</sub> P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Q <span style="font-family:Symbol;">Û</span> (P<span style="font-family:Symbol;">®</span> Q)<span style="font-family:Symbol;">Ù</span> (Q<span style="font-family:Symbol;">®</span> P)<br /><br />E<sub>22</sub> (P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Q) <span style="font-family:Symbol;">Û</span> (P<span style="font-family:Symbol;">Ù</span> Q) <span style="font-family:Symbol;">Ú</span> (<span style="font-family:Symbol;">ù</span> P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span> Q)<br /><br /><br /> <b> <br />Example 1:<br /> </b> <br />Show that R is a valid inference from the premises P<span style="font-family:Symbol;">®</span> Q, Q<span style="font-family:Symbol;">®</span> R, and P.<br /> <b> <br />Solution:<br /> </b> <br />{1} (1) P<span style="font-family:Symbol;">®</span> Q Rule P<br /> <br />{2} (2) P Rule P<br /> <br />{1,2} (3) Q Rule T, (1), (2) and I<sub>11</sub> (P, P<span style="font-family:Symbol;">®</span> Q <span style="font-family:Symbol;">Û</span> Q)<br /> <br />{4} (4) Q<span style="font-family:Symbol;">®</span> R Rule P<br /> <br />{1, 2, 4} (5) R Rule T, (3), (4) and I<sub>11<br /> </sub> <br />Therefore, R is a valid inference.<br /> <br /><br /> <b> <br />Example 2:<br /> </b> <br />Show that P <span style="font-family:Symbol;">Ú</span> Q follows logically from the premises C <span style="font-family:Symbol;">Ú</span> D, (C <span style="font-family:Symbol;">Ú</span> D) <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span> H,<br /> <br /><span style="font-family:Symbol;">ù</span> H<span style="font-family:Symbol;">®</span> (A <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>B) and (A <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>B) <span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ú</span> Q).<br /> <b> <br />Solution :<br /> </b> <br />{1} (1) (C <span style="font-family:Symbol;">Ú</span> D) Rule P<br /> <br />{2} (2) (C <span style="font-family:Symbol;">Ú</span> D)<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>H Rule P<br /><br />{1, 2} (3) <span style="font-family:Symbol;"> ù</span> H Rule T, (1), (2) and I<sub>11<br /> </sub><br />{4} (4) <span style="font-family:Symbol;"> ù </span>H<span style="font-family:Symbol;">®</span> (A <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>B) Rule P<br /> <br />{1, 2, 4} (5) A <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>B Rule T, (3), (4) and I<sub>11<br /> </sub> <br />{6} (6) (A <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>B)<span style="font-family:Symbol;">®</span> (P <span style="font-family:Symbol;">Ú</span> Q) Rule P<br /> <br />{1, 2, 4, 6} (<span style="font-family:Symbol;">7</span>) P <span style="font-family:Symbol;">Ú</span> Q Rule T, (5), (6) and I<sub>11<br /> </sub> <b> <br /><br /> <br />Example 3:<br /> </b> <br />Show that S<span style="font-family:Symbol;">Ú</span> R is tautologically implied by (P<span style="font-family:Symbol;">Ú</span> Q) <span style="font-family:Symbol;">Ù</span> (P<span style="font-family:Symbol;">®</span> R) <span style="font-family:Symbol;">Ù</span> (Q<span style="font-family:Symbol;">®</span> S).<br /> <b> <br />Solution:<br /> </b> <br />{1} (1) P <span style="font-family:Symbol;">Ú</span> Q Rule P <br />{1} (2) <span style="font-family:Symbol;">ù</span> P <span style="font-family:Symbol;">®</span> Q Rule T, (1), E<sub>1</sub> and E<sub>16</sub><br /> <br />{3} (3) Q<span style="font-family:Symbol;">®</span> S Rule P<br /> <br />{1,3} (4) <span style="font-family:Symbol;">ù</span> P<span style="font-family:Symbol;">®</span> S Rule T, (2), (3) and I<sub>13<br /> </sub> <br />{1,3} (5) <span style="font-family:Symbol;">ù</span> S<span style="font-family:Symbol;">®</span> P Rule T, (4) and E<sub>18<br /> </sub> <br />{6} (6) P<span style="font-family:Symbol;">®</span> R Rule P<br /> <br />{1,3,6} (7) <span style="font-family:Symbol;">ù</span> S<span style="font-family:Symbol;">®</span> R Rule T, (5), (6) and I<sub>13<br /> </sub> <br />{1,3,6} (8) S<span style="font-family:Symbol;">Ú</span> R Rule T, (7), E<sub>16</sub> and E<sub>1<br /> </sub> <b> <br /><br /> <br />Example 4:<br /> </b> <br />Show that (P<span style="font-family:Symbol;">Ú</span> Q) <span style="font-family:Symbol;">Ù</span> R is a valid conclusion from the premises P<span style="font-family:Symbol;">Ú</span> Q, Q<span style="font-family:Symbol;">®</span> R, P<span style="font-family:Symbol;">®</span> M and <span style="font-family:Symbol;">ù</span> M.<br /> <b> <br />Solution :<br /> </b> <br />{1} (1) <span style="font-family:Symbol;">ù</span> M Rule P<br /> <br />{2} (2) P<span style="font-family:Symbol;">®</span> M Rule P<br /> <br />{1,2} (3) <span style="font-family:Symbol;">ù</span> P Rule T, (1), (2) and I<sub>12<br /> </sub> <br />{4} (4) P<span style="font-family:Symbol;">Ú</span> Q Rule P<br /> <br />{1, 2, 4} (5) Q Rule T, (3), (4) and I<sub>10<br /> </sub> <br />{6} (6) Q<span style="font-family:Symbol;">®</span> R Rule P<br /> <br />{1, 2, 4, 6} (7) R Rule T, (5), (6) and I<sub>11<br /> </sub> <br />{1, 2, 4, 6} (8) (P <span style="font-family:Symbol;">Ú</span> Q)<span style="font-family:Symbol;">Ù</span> R Rule T, (4), (7) and I<sub>9<br /> </sub> <b> <br /><br /> <br />Example 5:<br /> </b> <br />Show that <span style="font-family:Symbol;">ù</span>P is a valid inference solution form <span style="font-family:Symbol;">ù</span>Q, P<span style="font-family:Symbol;">®</span> Q.<br /> <b> <br />Solution:<br /> </b> <br />{1} (1) P<span style="font-family:Symbol;">®</span> Q Rule P<br /> <br />{1} (2) <span style="font-family:Symbol;">ù</span>Q<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>P Rule T, (1) and E<sub>18<br /> </sub> <br />{3} (3) <span style="font-family:Symbol;">ù</span>Q Rule P<br /> <br />{1, 3} (4) <span style="font-family:Symbol;">ù</span>P Rule T, (2), (3), and I<sub>11<br /> </sub> <b> <br /><br /> <br />Example 6:<br /> </b> <br />Show that R is a valid inference from the premises P<span style="font-family:Symbol;">Ú</span> Q, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> R.<br /> <b> <br />Solution:<br /> </b> <br />{1} (1) P<span style="font-family:Symbol;">Ú</span> Q Rule P<br /> <br />{2} (2) Q<span style="font-family:Symbol;">®</span> R Rule P<br /> <br />{1, 2} (3) <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">®</span> R Rule T, (1), (2) and I<sub>13<br /> </sub> <br />{4} (4) P<span style="font-family:Symbol;">®</span> R Rule P<br /> <br />{1, 2} (5) <span style="font-family:Symbol;">ù</span>R<span style="font-family:Symbol;">®</span> P Rule T, (3) and E<sub>16<br /> </sub> <br />{1, 2, 4} (6) R Rule T, (4), (5) and I<sub>13<br /> </sub> <b> <br /><br /> <br />Rule CP: Rule of Conditional Proof.<br /> </b> <br />If we can derive S from R and a set of premises, then we can derive R<span style="font-family:Symbol;">®</span> S from the set of premises alone.<br /> <br />If the conclusion is of the form R<span style="font-family:Symbol;">®</span> S, then R is taken as an additional premise and S is derived from the given premises and R.<br /> <b> <br /><br /> <br />Example 7:<br /> </b> <br />Show that R<span style="font-family:Symbol;">®</span> S can be derived from the premises P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> S), <span style="font-family:Symbol;">ù</span>R<span style="font-family:Symbol;">Ú</span> P, and Q.<br /> <b> <br />Solution:<br /> </b> <br />We shall include R as an additional premise and show S first<br /> <br />{1} (1) <span style="font-family:Symbol;">ù</span>R<span style="font-family:Symbol;">Ú</span> P Rule P<br /> <br />{2} (2) R Rule P (assumed premise)<br /> <br />{1, 2} (3) P Rule T, (1), (2) and I<sub>10<br /> </sub> <br />{4} (4) P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> S) Rule P<br /> <br />{1, 2, 4} (5) Q<span style="font-family:Symbol;">®</span> S Rule T, (3), (4) and I<sub>11<br /> </sub> <br />{6} (6) Q Rule P<br /> <br />{1, 2, 4, 6} (7) S Rule T, (5), (6) and I<sub>11<br /> </sub> <br />{1, 2, 4, 6} (8) R<span style="font-family:Symbol;">®</span> S Rule CP<br /> <br /> </nobr><br /><span style="font-size:85%;"> </span></p><p align="right"><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Section1.3.htm#1.3%20Theory%20of%20Inference">Back to top</a><br /> </span><b> <br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Image/Sectio11.gif" width="775" border="0" height="10" /> </b> </p><p align="left"><b> <br /><br /> </b><b> <a name="1.3.4 Consistency of Premises and Indirect Method of Proof.">1.3.4 Consistency of Premises and Indirect Method of Proof.<br /> <br /> </a></b> <i>A set of formulas H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub> is said to be <b>consistent</b> if their conjunction has the truth value T for some assignment of the truth values to the atomic variables appearing in H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub>.</i><br /><b><br /></b><i>If, for every assignment of the truth values to the atomic variables, atleast one of the formulas H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub> is false, so that their conjunction is identically false, then the formulas H<sub>1</sub>,H<sub>2</sub>,…,H<sub>m</sub> are called<b> inconsistent</b></i>.<br /><br /><br /> A set of formulas H<sub>1</sub>, H<sub>2</sub>,…,H<sub>m</sub> are inconsistent if their conjunction implies a contradiction.<br /><br />That is, H<sub>1<span style="font-family:Symbol;">Ù</span> </sub>H<sub>2<span style="font-family:Symbol;">Ù</span> </sub>…<span style="font-family:Symbol;">Ù</span> H<sub>m</sub> <span style="font-family:Symbol;">Þ</span> R <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>R<br /><br />where R is any formula. R <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>R is a contradiction.<br /><br />In order to show that a conclusion C follows logically from the premises<br /><br />H<sub>1</sub>, H<sub>2</sub>, … , H<sub>m</sub> we assume that C is false and consider <span style="font-family:Symbol;">ù </span>C as an additional premises.<br /><b><br /><br /><nobr><br />Example 1:<br /></nobr></b><br />Show that <span style="font-family:Symbol;">ù</span>(P<span style="font-family:Symbol;">Ù</span> Q) follows from <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>Q.<br /><b><br />Solution:<br /></b><br />We introduce <span style="font-family:Symbol;">ù ù</span>(P<span style="font-family:Symbol;">Ù</span> Q) as an additional premise and show that this additional premise leads to contradiction.<br /><br />{1} (1) <span style="font-family:Symbol;">ù ù</span>(P<span style="font-family:Symbol;">Ù</span> Q) Rule P(assumed)<br /><br />{1} (2) P<span style="font-family:Symbol;">Ù</span> Q Rule T, (1), and E<sub>1<br /></sub><br />{1} (3) P Rule T, (2) and I<sub>1<br /></sub><br />{4} (4) <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>Q Rule P<br /><br />{4} (5) <span style="font-family:Symbol;">ù</span>P Rule T, (4) and I<sub>1<br /></sub><br />{1, 4} (6) P<span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>P Rule T, (3), (5) and I<sub>9<br /></sub><br /><br /><b><br />Example 2:<br /></b><br />Show that the given premises P<span style="font-family:Symbol;">®</span> Q, P<span style="font-family:Symbol;">®</span> R, Q<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R, P are inconsistent.<br /><b><br />Solution:<br /></b><br />{1} (1) P<span style="font-family:Symbol;">®</span> Q Rule P<br /><br />{2} (2) Q<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R Rule P<br /><br />{1, 2} (3) P<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R Rule T, (1), (2) and I<sub>13<br /></sub><br />{4} (4) P<span style="font-family:Symbol;">®</span> R Rule P<br /><br />{5} (5) P Rule P<br /><br />{4, 5} (6) R Rule T, (4), (5) and I<sub>11<br /></sub><br />{1, 2, 5} (7) <span style="font-family:Symbol;">ù</span>R Rule T, (3), (5), and I<sub>11<br /></sub><br />{1, 2, 4, 5} (8) R <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>R Rule T, (6), (7) and I<sub>9<br /></sub> <b><br /><br /><br />Example 3:<br /></b><br />Show the following premises are inconsistent<br /></p><ol><ol><li>If Jack misses many classes through illness, then he fails high school.</li><li>If Jack fails high school, then he is uneducated.</li><li>If Jack reads a lot of books, then he is not uneducated.</li><li>Jack misses many classes through illness and reads a lot of books.</li></ol></ol> <b><br />Solution :<br /></b><br /><b>P</b> : Jack misses many classes.<br /><br /><b>Q</b> : Jack fails high school.<br /><br /><b>R</b> : Jack reads a lot of books.<br /><br /><b>S</b> : Jack is uneducated.<br /><br /><br /><br />The premises are P <span style="font-family:Symbol;">®</span> Q, Q<span style="font-family:Symbol;">®</span> S, R<span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span> S, and P <span style="font-family:Symbol;">Ù</span> R.<br /><br />{1} (1) P <span style="font-family:Symbol;">®</span> Q Rule P<br /><br />{2} (2) Q<span style="font-family:Symbol;">®</span> S Rule P<br /><br />{1,2} (3) P <span style="font-family:Symbol;">®</span> S Rule T, (1), (2) and I<sub>13<br /></sub><br />{4} (4) R <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>S Rule P<br /><br />{4} (5) S <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R Rule T, (4) and E<sub>18<br /></sub><br />{1,2,4} (6) P <span style="font-family:Symbol;">®</span> <span style="font-family:Symbol;">ù</span>R Rule T, (3) (5) and I<sub>13<br /></sub><br />{1,2,4} (7) <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Úù</span>R Rule T, (5) and E<sub>16<br /></sub><br />{1,2,4} (8) <span style="font-family:Symbol;">ù</span>(P<span style="font-family:Symbol;">Ù</span> R) Rule T, (6) and E<sub>8<br /></sub><br />{9} (9) P<span style="font-family:Symbol;">Ù</span> R Rule P<br /><br />{1,2,4,9} (10) (P<span style="font-family:Symbol;">Ù</span> R) <span style="font-family:Symbol;">Ù</span> <span style="font-family:Symbol;">ù</span>(P<span style="font-family:Symbol;">Ù</span> R) Rule T, (8), (9) and I<sub>9<br /></sub><br /><br /><span style="font-size:85%;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.3/Section1.3.htm#1.3%20Theory%20of%20Inference">Back to top</a></span><br /><b><br /><img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Image/Sectio11.gif" width="767" border="0" height="10" /><br /><br /><br /><br /><span style="font-size:130%;"><a name="Exercise :">Exercise:</a></span><br /></b><br /><br /> <ol><li>Show that the conclusion C follows from the premises H<sub>1</sub>, H<sub>2</sub>, …… in the following cases.</li><br />(a) <b> H</b><sub><b>1</b> </sub>: P<span style="font-family:Symbol;">®</span> Q <b> </b><b>C</b> : P<span style="font-family:Symbol;">®</span> (P<span style="font-family:Symbol;">Ù</span> Q)<br /><br />(b) <b> H</b><sub><b>1</b> </sub>: <span style="font-family:Symbol;">ù</span>P<span style="font-family:Symbol;">Ú</span> Q <b> </b><b>H</b><sub><b>2</b> </sub>: <span style="font-family:Symbol;">ù</span>(Q<span style="font-family:Symbol;">Ù</span><span style="font-family:Symbol;">ù</span>R) <b> </b><b>H</b><sub><b>3</b> </sub>: <span style="font-family:Symbol;">ù</span>R <b>C </b>: <span style="font-family:Symbol;">ù</span>P<br /><br />(c) <b> H</b><sub><b>1</b> </sub>: <span style="font-family:Symbol;">ù</span>P <b> </b><b>H</b><sub><b>2</b> </sub>: P<span style="font-family:Symbol;">Ú</span> Q <b> </b><b>C</b> : Q<br /><br />(d) <b> H</b><sub><b>1</b> </sub>: <span style="font-family:Symbol;">ù</span>P <b> </b><b>H</b><sub><b>2</b> </sub>: P<span style="font-family:Symbol;">®</span> Q <b> </b><b>C</b> : <span style="font-family:Symbol;">ù</span>P<br /><br />(e) <b> H</b><sub><b>1</b> </sub>: P<span style="font-family:Symbol;">®</span> Q <b>H</b><sub><b>2</b> </sub>: Q<span style="font-family:Symbol;">®</span> R <b> </b><b>C</b> : P<span style="font-family:Symbol;">®</span> R<br /><br />(f) <b> H</b><sub><b>1</b> </sub>: R <b> </b><b>H</b><sub><b>2</b> </sub>: P<span style="font-family:Symbol;">Ú</span><span style="font-family:Symbol;">ù</span>P <b> </b><b>C</b> : R<br /><br /><li>Determine whether the conclusion C is valid in the following, where H<sub>1</sub>,H<sub>2</sub>,…… are the premises. </li></ol> (a) <b> H</b><sub><b>1</b> </sub>: P<span style="font-family:Symbol;">®</span> Q <b> </b> <b>H</b><sub><b>2</b> </sub>: <span style="font-family:Symbol;">ù</span>Q <b>C</b> : P<br /> <br /> (b) <b> H</b><sub><b>1</b> </sub>: P<span style="font-family:Symbol;">Ú</span> Q <b> </b> <b>H<sub>2</sub> </b> : P<span style="font-family:Symbol;">®</span> R <b> </b> <b>H<sub>3</sub> </b> : Q<span style="font-family:Symbol;">®</span> R <b> </b> <b>C</b> : R<br /> <br /> (c) <b> H<sub>1</sub> </b> : P<span style="font-family:Symbol;">®</span> (Q<span style="font-family:Symbol;">®</span> R) <b> </b> <b>H<sub>2</sub> </b> : P<span style="font-family:Symbol;">Ù</span> Q <b> </b> <b>C</b> : R<br /><br /> 3. Without constructing a truth table, show that A<span style="font-family:Symbol;">Ù</span> E is not a valid consequence of A<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> B,<br /> B <img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> (C<span style="font-family:Symbol;">Ù</span>D), C<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> (A<span style="font-family:Symbol;">Ú</span> E), A<span style="font-family:Symbol;">Ú</span> E. <br /> Also show that A<span style="font-family:Symbol;">Ú</span>C is not a valid consequence of A<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> (B<span style="font-family:Symbol;">®</span> C), B<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> (<span style="font-family:Symbol;">ù</span>A<span style="font-family:Symbol;">Úù</span>C), C<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> (A<span style="font-family:Symbol;">Úù</span>B), B.<br /> <br /><br /> 4. Show that A<span style="font-family:Symbol;">Ú</span>B follows from P<span style="font-family:Symbol;">Ù</span> Q<span style="font-family:Symbol;">Ù</span> R, (Q<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> R)<span style="font-family:Symbol;">®</span> (A<span style="font-family:Symbol;">Ú</span>B).<br /> <br /> 5. Show without constructing truth tables that the following statements cannot all be true simultaneously. <br /> <br /> (a) P<img src="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.1/Image/Sectio12.gif" width="23" border="0" height="16" /> Q Q<span style="font-family:Symbol;">®</span> R <span style="font-family:Symbol;">ù</span> R<span style="font-family:Symbol;">Ú</span>S <span style="font-family:Symbol;">ù</span> P<span style="font-family:Symbol;">®</span> S <span style="font-family:Symbol;">ù</span> S<br /> <br /> (b) R<span style="font-family:Symbol;">Ú</span>M <span style="font-family:Symbol;"> ù</span> R<span style="font-family:Symbol;">Ú</span>S <span style="font-family:Symbol;"> ù</span> M <span style="font-family:Symbol;"> ù</span> SSumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1tag:blogger.com,1999:blog-7399621945608861143.post-61519243867660886872008-08-12T04:18:00.001-07:002008-12-23T01:55:02.556-08:00Normal Forms<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///D:%5CDOCUME%7E1%5Cshesu04%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C06%5Cclip_filelist.xml"><link rel="Edit-Time-Data" href="file:///D:%5CDOCUME%7E1%5Cshesu04%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C06%5Cclip_editdata.mso"><!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--><title>© Moreniche</title><o:smarttagtype namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="place"></o:smarttagtype><!--[if gte mso 9]><xml> <o:documentproperties> <o:author>Marcus Polo</o:Author> <o:version>11.9999</o:Version> </o:DocumentProperties> </xml><![endif]--><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> <w:usefelayout/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><!--[if !mso]><object classid="clsid:38481807-CA0E-42D2-BF39-B33AF135CC4D" id="ieooui"></object> <style> st1\:*{behavior:url(#ieooui) } </style> <![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:SimSun; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:ËÎÌå; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:"\@SimSun"; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:"\@Arial Unicode MS"; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:SimSun;} a:link, span.MsoHyperlink {color:blue; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {color:blue; text-decoration:underline; text-underline:single;} p {mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:SimSun;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:95832565; mso-list-template-ids:443430514;} @list l0:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l0:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l1 {mso-list-id:309605048; mso-list-template-ids:-202708646;} @list l1:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l2 {mso-list-id:426849391; mso-list-template-ids:1952068542;} @list l3 {mso-list-id:565576782; mso-list-template-ids:-1110802726;} @list l3:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l4 {mso-list-id:583227614; mso-list-template-ids:607798466;} @list l4:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l5 {mso-list-id:890120939; mso-list-template-ids:1547965002;} @list l5:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l5:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l6 {mso-list-id:1339888043; mso-list-template-ids:-542727176;} @list l6:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l6:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l7 {mso-list-id:1426539645; mso-list-template-ids:1875511450;} @list l7:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l7:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l8 {mso-list-id:1854954640; mso-list-template-ids:1911345182;} @list l8:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l8:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l9 {mso-list-id:1947032497; mso-list-template-ids:-912366840;} @list l9:level1 {mso-level-number-format:alpha-lower; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l9:level2 {mso-level-number-format:alpha-lower; mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> <p><a name="Normal_Forms"></a><o:p></o:p></p> <p>Let A(P<sub>1</sub>, P<sub>2</sub>, P<sub>3</sub>, …, P<sub>n</sub>) be a statement formula where P<sub>1</sub>, P<sub>2</sub>, P<sub>3</sub>, …, P<sub>n</sub> are the atomic variables. If A has truth value T for all possible assignments of the truth values to the variables P<sub>1</sub>, P<sub>2</sub>, P<sub>3</sub>, …, P<sub>n</sub> , then A is said to be a tautology. If A has truth value F, then A is said to be identically false or a contradiction.<o:p></o:p></p> <p> <o:p></o:p></p> <p><a name="1.2.1_Disjunctive_Normal_Forms"><b>1.2.1 Disjunctive </b></a><st1:place st="on"><span style=""><b>Normal</b></span></st1:place><span style=""><b> Forms</b></span><o:p></o:p></p> <p><i>A product of the variables and their negations in a formula is called an <b>elementary product</b>. A sum of the variables and their negations is called an <b>elementary sum</b>. That is, a sum of elementary products is called a <b>disjunctive normal form</b> of the given formula.<o:p></o:p></i></p> <p> <o:p></o:p></p> <p><b>Example:<o:p></o:p></b></p> <p class="MsoNormal"><nobr>The disjunctive normal form of</nobr>
<br /> <!--[if !supportLineBreakNewLine]-->
<br /> <!--[endif]--><o:p></o:p></p> <nobr> <ol start="1" type="1"><li class="MsoNormal" style="">P<span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">®</span> Q) <span style="font-family: Symbol;">Û</span> P<span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Û</span> (P<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P)<span style="font-family: Symbol;">Ú</span> (P<span style="font-family: Symbol;">Ù</span> Q).<o:p></o:p></li><li class="MsoNormal" style=""><span style="font-family: Symbol;">ù</span> (P <span style="font-family: Symbol;">Ú</span> Q)<!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" spt="75" preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:formulas> <v:path extrusionok="f" gradientshapeok="t" connecttype="rect"> <o:lock ext="edit" aspectratio="t"> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1025" width="23" border="0" height="16" /><!--[endif]--> (P<span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> (P<span style="font-family: Symbol;">Ú</span> Q)<span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ù</span> Q))<span style="font-family: Symbol;">Ú</span> ((P<span style="font-family: Symbol;">Ú</span> Q)<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> (P<span style="font-family: Symbol;">Ù</span> Q))
<br /> <span style="font-family: Symbol;"> Û</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P) <span style="font-family: Symbol;">Ú</span> (Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q). <o:p></o:p></li></ol> </nobr> <p style="text-align: right;" align="right"><span style="font-size: 10pt;"> </span><o:p></o:p></p> <p style="text-align: right;" align="right"><span style="font-size: 10pt;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Section1.2.htm#1.2%20%20%20%20Normal%20Forms">Back to top</a></span><o:p></o:p></p> <p><span style="font-size: 10pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1026" type="#_x0000_t75" alt="" style="'width:579pt;height:7.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image003.jpg" shapes="_x0000_i1026" width="772" border="0" height="10" /><!--[endif]--></span><o:p></o:p></p> <p><b> </b><o:p></o:p></p> <p><a name="1.2.2_Conjunctive_Normal_Forms"><b>1.2.2 Conjunctive </b></a><st1:place st="on"><span style=""><b>Normal</b></span></st1:place><span style=""><b> Forms</b></span><o:p></o:p></p> <p><i>A formula which is equivalent to a given formula and which consists of a product of elementary sums is called a <b>conjunctive normal form</b> of a given formula.</i><o:p></o:p></p> <p> <o:p></o:p></p> <p><b>Example 1:<o:p></o:p></b></p> <p class="MsoNormal" style="margin-bottom: 12pt;"><nobr></nobr><nobr>The conjunctive normal form of
<br />
<br /></nobr><nobr>(a) P <span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">®</span> Q) <span style="font-family: Symbol;">Û</span> P <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q).
<br />
<br />(b) <span style="font-family: Symbol;">ù </span>(P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Û</span> <span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù </span>Q.
<br />
<br />(c) <span style="font-family: Symbol;"> ù</span> (P<!--[if gte vml 1]><v:shape id="_x0000_i1027" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1027" width="23" border="0" height="16" /><!--[endif]--> Q) <span style="font-family: Symbol;">Û</span> <span style="font-family: Symbol;">ù</span> ((P<span style="font-family: Symbol;">®</span> Q) <span style="font-family: Symbol;">Ù</span> (Q<span style="font-family: Symbol;">®</span> P))
<br /><span style="font-family: Symbol;"> Û</span> <span style="font-family: Symbol;">ù </span>((<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> P))
<br /><span style="font-family: Symbol;"> Û</span> <span style="font-family: Symbol;">ù </span>(<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> (<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> P)
<br /><span style="font-family: Symbol;"> Û</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P)
<br /><span style="font-family: Symbol;"> Û</span> (P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Ù</span> (Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> P) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q).
<br />
<br />(d) <span style="font-family: Symbol;">ù</span> (P <span style="font-family: Symbol;">Ú</span> Q) <!--[if gte vml 1]><v:shape id="_x0000_i1028" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1028" width="23" border="0" height="16" /><!--[endif]-->(P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> (P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">®</span> (P <span style="font-family: Symbol;">Ù</span> Q)) <span style="font-family: Symbol;">Ù</span> ((P <span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">®</span> <span style="font-family: Symbol;">ù</span> (P <span style="font-family: Symbol;">Ú</span> Q))
<br /><span style="font-family: Symbol;"> Û</span> ((P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q)) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> (P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q)
<br /><span style="font-family: Symbol;"> Û</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> P) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> P) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q).</nobr><b><o:p></o:p></b></p> <p><b>Example 2:<o:p></o:p></b></p> <p class="MsoNormal"><nobr>(a) Show that the formula <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ù</span> B) <span style="font-family: Symbol;">Ú</span> (A <span style="font-family: Symbol;">Ù</span> B) is a tautology.<span style="font-family: Symbol;">
<br />
<br /> ù</span> B <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ù</span> B) <span style="font-family: Symbol;">Ú</span> (A <span style="font-family: Symbol;">Ù</span> B) <span style="font-family: Symbol;">Û</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> ((<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> A) <span style="font-family: Symbol;">Ù</span> B)<span style="font-family: Symbol;">
<br /> Û</span> (<span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> A) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> B)<span style="font-family: Symbol;">
<br /> Û</span> T <span style="font-family: Symbol;">Ù</span> T<span style="font-family: Symbol;">
<br /> Û</span> T.
<br /> Therefore, it is a tautology.
<br />
<br />(b) Show that the formula (<span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ù</span> A) <span style="font-family: Symbol;">Ù</span> B is a contradiction.
<br />
<br /> (<span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ù</span> A) <span style="font-family: Symbol;">Ù</span> B <span style="font-family: Symbol;">Û</span> A <span style="font-family: Symbol;">Ù</span> (B <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> B)
<br /> <span style="font-family: Symbol;"> Û</span> A <span style="font-family: Symbol;">Ù</span> F<span style="font-family: Symbol;">
<br /></span> <span style="font-family: Symbol;"> Û</span> F.
<br /> Therefore, it is a contradiction.
<br />
<br />(c) Show that (Q<span style="font-family: Symbol;">®</span> P) <span style="font-family: Symbol;">Ù</span> ( <span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q) is a contradiction.
<br />
<br /> (Q<span style="font-family: Symbol;">®</span> P) <span style="font-family: Symbol;">Ù</span> ( <span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> P) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q)
<br /><span style="font-family: Symbol;"> Û </span>(<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û </span>F <span style="font-family: Symbol;">Ú</span> F<span style="font-family: Symbol;">
<br /> Û </span>F.
<br /> Therefore, it is a contradiction. </nobr><span style="font-size: 10pt;"><o:p></o:p></span></p> <p style="text-align: right;" align="right"><span style="font-size: 10pt;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Section1.2.htm#1.2%20%20%20%20Normal%20Forms">Back to top</a><o:p></o:p></span></p> <p><span style="font-size: 10pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1029" type="#_x0000_t75" alt="" style="'width:585.75pt;height:7.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image004.jpg" shapes="_x0000_i1029" width="781" border="0" height="10" /><!--[endif]--></span><o:p></o:p></p> <p> <o:p></o:p></p> <p><a name="1.2.3_Principal_Disjunctive_Normal_Form_"><b>1.2.3 Principal Disjunctive Normal Form (PDNF)</b></a><b><o:p></o:p></b></p> <p>Let us assume A and B be two statement variables. All possible formulas by using conjunction are given as follows. The total number of formulas for two variables A and B are 2<sup>2</sup> formulas. They are A <span style="font-family: Symbol;">Ù</span> B, A <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span>B,
<br /><span style="font-family: Symbol;">ù</span>A <span style="font-family: Symbol;">Ù</span> B and <span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> B.<o:p></o:p></p> <p>These are called minterms or Boolean conjunctions of A and B. The minterms (2<sup>n</sup> terms) are denoted by M<sub>0</sub>, M<sub>1</sub>, … ,M<sub>2</sub><sup><span style="font-size: 7.5pt;">n</span></sup><sub>-1</sub>.<o:p></o:p></p> <p><i>A formula equivalent to a given formula consisting of the disjunction of minterms only is called <b>PDNF</b> of the given formula.<o:p></o:p></i></p> <p> <o:p></o:p></p> <p><b>Example 1:<o:p></o:p></b></p> <p class="MsoNormal"><nobr>Obtain the PDNF of (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q)<span style="font-family: Symbol;">®</span> (P<!--[if gte vml 1]><v:shape id="_x0000_i1030" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1030" width="23" border="0" height="16" /><!--[endif]--> <span style="font-family: Symbol;">ù</span> Q)</nobr> <o:p></o:p></p> <p class="MsoNormal" style="text-align: center;" align="center"><o:p> </o:p></p> <div align="center"> <table class="MsoNormalTable" style="width: 350.25pt;" width="467" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 29.25pt;" valign="top" width="39"> <p style="text-align: center;" align="center"><b>P</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 21.75pt;" valign="top" width="29"> <p style="text-align: center;" align="center"><b>Q</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 49.5pt;" valign="top" width="66"> <p style="text-align: center;" align="center"><b><span style="font-family: Symbol;">ù</span> P</b><b><span style="font-family: Symbol;">Ú</span> </b><b><span style="font-family: Symbol;">ù</span> Q</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 48.75pt;" valign="top" width="65"> <p style="text-align: center;" align="center"><b>P <!--[if gte vml 1]><v:shape id="_x0000_i1031" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1031" width="23" border="0" height="16" /><!--[endif]--></b><b><span style="font-family: Symbol;">ù</span> Q</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 130.5pt;" valign="top" width="174"> <p style="text-align: center;" align="center"><b>(</b><b><span style="font-family: Symbol;">ù</span> P</b><b><span style="font-family: Symbol;">Ú</span> </b><b><span style="font-family: Symbol;">ù</span> Q) </b><b><span style="font-family: Symbol;">®</span> (P<!--[if gte vml 1]><v:shape id="_x0000_i1032" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1032" width="23" border="0" height="16" /><!--[endif]--> </b><b><span style="font-family: Symbol;">ù</span> Q)</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 29.25pt;" valign="top" width="39"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 21.75pt;" valign="top" width="29"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 49.5pt;" valign="top" width="66"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 48.75pt;" valign="top" width="65"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 130.5pt;" valign="top" width="174"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 29.25pt;" valign="top" width="39"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 21.75pt;" valign="top" width="29"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 49.5pt;" valign="top" width="66"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 48.75pt;" valign="top" width="65"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 130.5pt;" valign="top" width="174"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 29.25pt;" valign="top" width="39"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 21.75pt;" valign="top" width="29"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 49.5pt;" valign="top" width="66"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 48.75pt;" valign="top" width="65"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 130.5pt;" valign="top" width="174"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 29.25pt;" valign="top" width="39"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 21.75pt;" valign="top" width="29"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 49.5pt;" valign="top" width="66"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 48.75pt;" valign="top" width="65"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 130.5pt;" valign="top" width="174"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p> <o:p></o:p></p> <p class="MsoNormal"><nobr>From the above table
<br />
<br />(<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q)<span style="font-family: Symbol;">®</span> (P<!--[if gte vml 1]><v:shape id="_x0000_i1033" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1033" width="23" border="0" height="16" /><!--[endif]--> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Û</span> (P<span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P<span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> <!--[if gte vml 1]><v:shape id="_x0000_i1034" type="#_x0000_t75" alt="" style="'width:48.75pt;height:19.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image005.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image97.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image005.gif" shapes="_x0000_i1034" width="65" align="absmiddle" border="0" height="26" /><!--[endif]--><b><o:p></o:p></b></nobr></p> <p><b> <o:p></o:p></b></p> <p><b>Example 2:<o:p></o:p></b></p> <p>Obtain the PDNF of <!--[if gte vml 1]><v:shape id="_x0000_i1035" type="#_x0000_t75" alt="" style="'width:36.75pt;height:15.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image98.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1035" width="49" align="absmiddle" border="0" height="21" /><!--[endif]--><o:p></o:p></p> <div align="center"> <table class="MsoNormalTable" style="width: 130.5pt;" width="174" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center"><b>P</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center;" align="center"><b>Q</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 41%;" valign="top" width="41%"> <p style="text-align: center;" align="center"><b>P </b><b><span style="font-family: Symbol;">®</span> Q</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 41%;" valign="top" width="41%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 41%;" valign="top" width="41%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 41%;" valign="top" width="41%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 41%;" valign="top" width="41%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> <nobr></nobr></div> <p class="MsoNormal">
<br />P<span style="font-family: Symbol;">®</span> Q <span style="font-family: Symbol;">Û</span> (P<span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q)<span style="font-family: Symbol;">
<br /> Û </span>(<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P<span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û </span><!--[if gte vml 1]><v:shape id="_x0000_i1036" type="#_x0000_t75" alt="" style="'width:46.5pt;height:19.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image007.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image99.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image007.gif" shapes="_x0000_i1036" width="62" align="absmiddle" border="0" height="26" /><!--[endif]--><o:p></o:p></p> <p><b> </b><o:p></o:p></p> <p><b>Procedure for obtaining PDNF</b><o:p></o:p></p> <p class="MsoNormal"><b><nobr>Step1 :</nobr></b> Get rid of <!--[if gte vml 1]><v:shape id="_x0000_i1037" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1037" width="23" border="0" height="16" /><!--[endif]-->, <span style="font-family: Symbol;">®</span> .<b>
<br />Step2 :</b> Use De Morgan’s law.<b>
<br />Step3 : </b>Use distributive law.<b>
<br />Step4 : </b>Introduce missing factor in the disjunction.<b>
<br />Step5 : </b>If there is elementary product of the form.<o:p></o:p></p> <p>(P<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P) <span style="font-family: Symbol;">Ú</span> ( ) <span style="font-family: Symbol;">Ú</span> …… , delete P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> P.<o:p></o:p></p> <p> <o:p></o:p></p> <p>For every truth value T in the truth table of the given formula, select minterm which also has the value T for the same combination of the truth values of P and Q. The disjunction of these minterms will be equivalent to the given formula.<o:p></o:p></p> <p>To find a particular minterm M<sub>i</sub>, the subscript i is expressed in the binary representation and suitable number of 0’s are added to the left.<o:p></o:p></p> <p> <o:p></o:p></p> <p><b>Example 3:</b><o:p></o:p></p> <p>Obtain PDNF for <!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75" alt="" style="'width:30.75pt;height:15.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image008.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image100.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image008.gif" shapes="_x0000_i1038" width="41" align="absmiddle" border="0" height="21" /><!--[endif]-->.<o:p></o:p></p> <p><!--[if gte vml 1]><v:shape id="_x0000_i1039" type="#_x0000_t75" alt="" style="'width:223.5pt;height:99pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image009.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image101.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image009.gif" shapes="_x0000_i1039" width="298" border="0" height="132" /><!--[endif]--><o:p></o:p></p> <p>Assign 1 for P and 0 for <span style="font-family: Symbol;">ù</span> P.<o:p></o:p></p> <p><b> <o:p></o:p></b></p> <p><b>Example 4:<o:p></o:p></b></p> <p>Obtain PDNF for <!--[if gte vml 1]><v:shape id="_x0000_i1040" type="#_x0000_t75" alt="" style="'width:39pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image010.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image102.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image010.gif" shapes="_x0000_i1040" width="52" align="absmiddle" border="0" height="25" /><!--[endif]-->.<o:p></o:p></p> <p><b>Solution:<o:p></o:p></b></p> <p><!--[if gte vml 1]><v:shape id="_x0000_i1041" type="#_x0000_t75" alt="" style="'width:247.5pt;height:99pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image011.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image103.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image011.gif" shapes="_x0000_i1041" width="330" border="0" height="132" /><!--[endif]--><o:p></o:p></p> <p> <o:p></o:p></p> <p><b>Example 5:<o:p></o:p></b></p> <p class="MsoNormal"><nobr>Obtain PDNF for P<span style="font-family: Symbol;">®</span> ((P<span style="font-family: Symbol;">®</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> (<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> P))).
<br />
<br /></nobr><b><nobr>Solution:
<br />
<br /></nobr></b>P<span style="font-family: Symbol;">®</span> ((P<span style="font-family: Symbol;">®</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> (<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> P))) <span style="font-family: Symbol;">Û</span> P<span style="font-family: Symbol;">®</span> ((P<span style="font-family: Symbol;">®</span> Q <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ù</span> Q)))<span style="font-family: Symbol;">
<br /> Û</span> P<span style="font-family: Symbol;">®</span> ((P<span style="font-family: Symbol;">®</span> P <span style="font-family: Symbol;">Ù</span> Q))<span style="font-family: Symbol;">
<br /> Û</span> P<span style="font-family: Symbol;">®</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q))<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q))<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> (Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q)) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> <!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75" alt="" style="'width:46.5pt;height:19.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image104.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1042" width="62" align="absmiddle" border="0" height="26" /><!--[endif]-->.<b><o:p></o:p></b></p> <p><b> <o:p></o:p></b></p> <p style="text-align: right;" align="right"><span style="font-size: 10pt;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Section1.2.htm#1.2%20%20%20%20Normal%20Forms">Back to top</a></span><o:p></o:p></p> <p><span style="font-size: 10pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1043" type="#_x0000_t75" alt="" style="'width:8in;height:7.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.jpg" shapes="_x0000_i1043" width="768" border="0" height="10" /><!--[endif]--></span><o:p></o:p></p> <p style="text-align: right;" align="right"><b> </b><o:p></o:p></p> <p><a name="1.2.4_Principal_Conjunctive_Normal_Forms"><b>1.2.4 Principal Conjunctive Normal Forms (PCNF)</b></a><o:p></o:p></p> <p><i>The duals of minterms are called maxterms</i>. For a given number of variables the maxterm consists of disjunctions in which each variable or its negation, but not both, appears only once.<o:p></o:p></p> <p><i>For a given formula, an equivalent formula consisting of conjunctions of maxterms only is known as <b>its principal conjunctive normal form</b>. This is also called the <b>product of sums canonical form</b>.<o:p></o:p></i></p> <p> <o:p></o:p></p> <p><b>Example 1:<o:p></o:p></b></p> <p class="MsoNormal"><nobr>Obtain the PCNF of (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">®</span> R) <span style="font-family: Symbol;">Ù</span> (Q<!--[if gte vml 1]><v:shape id="_x0000_i1044" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1044" width="23" border="0" height="16" /><!--[endif]--> P).<b>
<br />
<br />Solution:
<br />
<br /></b>(<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">®</span> R) <span style="font-family: Symbol;">Ù</span> (Q<!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1045" width="23" border="0" height="16" /><!--[endif]--> P) <span style="font-family: Symbol;">Û</span> (P <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> ((Q<span style="font-family: Symbol;">®</span> P) <span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">®</span> Q))
<br /> <span style="font-family: Symbol;"> Û </span>(P <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> ((<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> P) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q))<span style="font-family: Symbol;">
<br /> Û </span>(P <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> (Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q)) <span style="font-family: Symbol;">Ù</span> ((P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> (R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R)) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> (R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R))<span style="font-family: Symbol;">
<br /> Û </span>(P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R)
<br />
<br /> S <span style="font-family: Symbol;">Û</span> <span style="font-family: Symbol;">p</span> (0,2,3,4,5).<span style="font-family: Symbol;">
<br />
<br />ù</span> S <span style="font-family: Symbol;">Û</span> consisting of missing maxterms<span style="font-family: Symbol;">
<br /> Û</span> M<sub>1</sub> <span style="font-family: Symbol;">Ù</span> M<sub>6</sub> <span style="font-family: Symbol;">Ù</span> M<sub>7</sub><span style="font-family: Symbol;">
<br /> Û </span>(P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R).
<br /><b>
<br />
<br />Example2:
<br />
<br /></b>Obtain PCNF for A : (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">®</span> R) <span style="font-family: Symbol;">Ù</span> ((Q<span style="font-family: Symbol;">®</span> P) <span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">®</span> Q)).<b>
<br />
<br />Solution:
<br />
<br /></b>A <span style="font-family: Symbol;">Û</span> (P<span style="font-family: Symbol;">Ú</span> R)<span style="font-family: Symbol;">Ù</span> ((<span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> P)<span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q))<span style="font-family: Symbol;">
<br /> Û</span> (P<span style="font-family: Symbol;">Ú</span> R<span style="font-family: Symbol;">Ú</span> (Q<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q)) <span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> (R<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R)) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> (R<span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R))<span style="font-family: Symbol;">
<br /> Û</span> (P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> R)<span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> R)<span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> R)<span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R)<span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> R)<span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R)<span style="font-family: Symbol;">
<br /> Û</span> (P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (P<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">Ú</span> Q<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R)<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span> (0,2,3,4,5).</nobr> <b><o:p></o:p></b></p> <p><b> <o:p></o:p></b></p> <p><b>Example 3:<o:p></o:p></b></p> <p>From the given truth table formula S, determine its PDNF and PCNF<o:p></o:p></p> <p style="text-align: center;" align="center"><b>Table 1.2.4<o:p></o:p></b></p> <div align="center"> <table class="MsoNormalTable" style="width: 196.5pt;" width="262" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center"><b>C</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center"><b>S</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 24%;" valign="top" width="24%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 25%;" valign="top" width="25%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 22%;" valign="top" width="22%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 29%;" valign="top" width="29%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p class="MsoNormal">
<br /><nobr>By choosing the minterms corresponding to each T value of S,
<br />
<br />the PDNF of S <!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75" alt="" style="'width:148.5pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image105.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1046" width="198" align="absmiddle" border="0" height="25" /><!--[endif]-->.
<br />
<br />By choosing the maxterms corresponding to each F value of S,
<br />
<br />the PCNF of S <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> C) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> B <span style="font-family: Symbol;">Ú ù</span> C) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> B <span style="font-family: Symbol;">Ú</span> C) <span style="font-family: Symbol;">Ù</span> (A <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> C) <span style="font-family: Symbol;">Ù</span> (A <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> C)<span style="font-family: Symbol;">
<br /> Ù</span> (A <span style="font-family: Symbol;">Ú</span> B <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> C) .
<br />
<br />
<br /><b>Example 4:
<br />
<br /></b>Form the given truth table formula S, determine its PDNF and PCNF</nobr><o:p></o:p></p> <div align="center"> <table class="MsoNormalTable" style="width: 183pt;" width="244" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center"><b>C</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center"><b>S</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 27%;" valign="top" width="27%"> <p style="text-align: center;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 23%;" valign="top" width="23%"> <p style="text-align: center;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p class="MsoNormal">
<br /><nobr>By choosing minterms corresponding to each T value of S,
<br />
<br />the PDNF of S <span style="font-family: Symbol;">Û</span> (A <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ù</span> C) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ù</span> B <span style="font-family: Symbol;">Ù</span> C) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ù</span> B <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> C) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> C)<span style="font-family: Symbol;">
<br /> Û</span> <!--[if gte vml 1]><v:shape id="_x0000_i1047" type="#_x0000_t75" alt="" style="'width:57.75pt;height:19.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image015.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image106.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image015.gif" shapes="_x0000_i1047" width="77" align="absmiddle" border="0" height="26" /><!--[endif]-->.
<br />
<br />By choosing maxterms corresponding to each F value of S,
<br />
<br />the PCNF of S <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> C) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> B <span style="font-family: Symbol;">Ú</span> C) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> A <span style="font-family: Symbol;">Ú</span> B <span style="font-family: Symbol;">Ú</span> C) <span style="font-family: Symbol;">Ù</span> (A <span style="font-family: Symbol;">Ú</span> B <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> C)
<br /> <span style="font-family: Symbol;">Û</span> <span style="font-family: Symbol;">p</span>(1,4,6,7).
<br /><b>
<br />
<br />Example 5:
<br />
<br /></b>Obtain the product of sums canonical form of the formula A which is given by
<br />(P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R)<b>
<br />
<br />Solution:</b><span style="font-family: Symbol;">
<br />
<br />ù</span> A <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R)<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span>(0,3,7).<span style="font-family: Symbol;">
<br />
<br />ù</span> (<span style="font-family: Symbol;">ù</span> A) <span style="font-family: Symbol;">Û</span> consisting of missing maxterms<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span>(1,2,4,5,6)<span style="font-family: Symbol;">
<br /> Û</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R) .
<br /><b>
<br />
<br />Example 6:
<br />
<br /></b>Obtain the product-of-sums canonical form of the formula A, which is given by
<br />(<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ù</span> S) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ù</span> S) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> S).<b>
<br />
<br />Solution:</b><span style="font-family: Symbol;">
<br />
<br />ù</span> A <span style="font-family: Symbol;">Û</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú ù</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span>
<br /> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> S)<span style="font-family: Symbol;">
<br /> Û</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span>
<br /> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> S)<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span>(5, 6, 9, 10, 12).<span style="font-family: Symbol;">
<br />
<br />ù</span> (<span style="font-family: Symbol;">ù</span> A) <span style="font-family: Symbol;">Û</span> consisting of missing maxterms<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span>(0,1,2,3,4,7,8,11,13,14,15)<span style="font-family: Symbol;">
<br /> Û</span> M<sub>0</sub><span style="font-family: Symbol;">Ù</span> M<sub>1</sub><span style="font-family: Symbol;">Ù</span> M<sub>2</sub><span style="font-family: Symbol;">Ù</span> M<sub>3</sub><span style="font-family: Symbol;">Ù</span> M<sub>4</sub><span style="font-family: Symbol;">Ù</span> M<sub>7</sub><span style="font-family: Symbol;">Ù</span> M<sub>8</sub><span style="font-family: Symbol;">Ù</span> M<sub>11</sub><span style="font-family: Symbol;">Ù</span> M<sub>13</sub><span style="font-family: Symbol;">Ù</span> M<sub>14</sub><span style="font-family: Symbol;">Ù</span> M<sub>15
<br /></sub><span style="font-family: Symbol;"> Û</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> S)
<br /> <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ù</span>
<br /> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> S) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> S).
<br /><b>
<br />
<br />Example 7:
<br />
<br /></b>Obtain the product of sums canonical form of (P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q).<b>
<br />
<br />Solution:</b><span style="font-family: Symbol;">
<br />
<br />ù</span> A <span style="font-family: Symbol;">Û</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> Q) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q)<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span>(1,2,3).<span style="font-family: Symbol;">
<br />
<br />ù</span> (<span style="font-family: Symbol;">ù</span> A) <span style="font-family: Symbol;">Û</span> consisting of missing maxterms<span style="font-family: Symbol;">
<br /> Û</span> <span style="font-family: Symbol;">p</span>(0)<span style="font-family: Symbol;">
<br /> Û</span> M<sub>0</sub><span style="font-family: Symbol;">
<br /> Û</span> P <span style="font-family: Symbol;">Ú</span> Q.</nobr>
<br /><span style="font-size: 10pt;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Section1.2.htm#1.2%20%20%20%20Normal%20Forms">
<br /></a></span><o:p></o:p></p> <p style="text-align: right;" align="right"><span style="font-size: 10pt;"><a href="file:///D:/Documents%20and%20Settings/shesu04/Desktop/dms/Sethuraman%20-%20MA109/Unit1/Section1.2/Section1.2.htm#1.2%20%20%20%20Normal%20Forms">Back to top</a></span> <o:p></o:p></p> <p><span style="font-size: 10pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1048" type="#_x0000_t75" alt="" style="'width:8in;height:7.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.jpg" shapes="_x0000_i1048" width="768" border="0" height="10" /><!--[endif]--></span><o:p></o:p></p> <p> <o:p></o:p></p> <p><a name="Exercise:"><b><span style="font-size: 13.5pt;">Exercise:</span></b></a><b><o:p></o:p></b></p> <p>1. Write the following statements in symbolic form using the statements<o:p></o:p></p> <p><b> P</b> : Sandeep is rich. <b>Q</b> : Sandeep is happy.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>Sandeep is poor but happy.<o:p></o:p></nobr></li><li class="MsoNormal" style="">Sandeep is rich or unhappy.<o:p></o:p></li><li class="MsoNormal" style="">Sandeep is neither rich nor happy.<o:p></o:p></li><li class="MsoNormal" style="">Sandeep is poor or he is both rich and unhappy.<o:p></o:p></li></ol></ol> <p>2. Write the negation of each of the following statements as simply as possible.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>He is tall but handsome.<o:p></o:p></nobr></li><li class="MsoNormal" style="">He has blond hair or blue eyes.<o:p></o:p></li><li class="MsoNormal" style="">He is neither rich nor happy.<o:p></o:p></li><li class="MsoNormal" style="">He lost his job or he did not go to work today.<o:p></o:p></li><li class="MsoNormal" style="">Neither Siva nor Sriram is unhappy.<o:p></o:p></li><li class="MsoNormal" style="">Ajay speaks Bengali or Hindi, but not German.<o:p></o:p></li><li class="MsoNormal" style="">If he studies, he will pass the exam.<o:p></o:p></li><li class="MsoNormal" style="">He swims if and only if the water is warm.<o:p></o:p></li><li class="MsoNormal" style="">If it snows, then he does not drive the car.<o:p></o:p></li><li class="MsoNormal" style="">If it is cold, then he wears a coat but no sweater.<o:p></o:p></li><li class="MsoNormal" style="">If he studies then he will go to college or to art school.<o:p></o:p></li></ol></ol> <p>3. Write the following statement in symbolic form using the statements<o:p></o:p></p> <p><b> P</b>: It is cold. <b>Q</b>: It rains.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>It rains only if it is cold.<o:p></o:p></nobr></li><li class="MsoNormal" style="">A necessary condition for it to be cold is that it rain.<o:p></o:p></li><li class="MsoNormal" style="">A sufficient condition for it to be cold is that it rain.<o:p></o:p></li><li class="MsoNormal" style="">Whenever it rains it is cold.<o:p></o:p></li><li class="MsoNormal" style="">It never rains when it is cold.<o:p></o:p></li></ol></ol> <p>4. Write each statement in symbolic form using <o:p></o:p></p> <p> <b>P</b> : He is rich <b>Q</b> : He is happy.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>If he is rich then he is unhappy.<o:p></o:p></nobr></li><li class="MsoNormal" style="">He is neither rich nor happy.<o:p></o:p></li><li class="MsoNormal" style="">It is necessary to be poor in order to be happy.<o:p></o:p></li><li class="MsoNormal" style="">To be poor is to be unhappy.<o:p></o:p></li><li class="MsoNormal" style="">Being rich is a sufficient condition to being happy.<o:p></o:p></li><li class="MsoNormal" style="">Being rich is a necessary condition to being happy.<o:p></o:p></li><li class="MsoNormal" style="">One is never happy when one is rich.<o:p></o:p></li><li class="MsoNormal" style="">He is poor only if he is happy.<o:p></o:p></li><li class="MsoNormal" style="">To be rich means the same as to be happy.<o:p></o:p></li><li class="MsoNormal" style="">He is poor or else he is both rich and happy.<o:p></o:p></li></ol></ol> <p>5. Write the negation of each statement in as simple a sentence as possible.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>If Sindhu is a poet, then she is poor.<o:p></o:p></nobr></li><li class="MsoNormal" style="">If Madhu passed the test, then he studied.<o:p></o:p></li><li class="MsoNormal" style="">If x is less than zero, then x is not positive.<o:p></o:p></li><li class="MsoNormal" style="">If it is cold, he wears a hat.<o:p></o:p></li><li class="MsoNormal" style="">If productivity increases, then wages rise.<o:p></o:p></li></ol></ol> <p>6. From the formulas given below select those which are well formed and indicate which ones are tautologies
<br /> or contradictions.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>(A <span style="font-family: Symbol;">® </span>( A <span style="font-family: Symbol;">Ú </span>Q))<o:p></o:p></nobr></li><li class="MsoNormal" style="">((A <span style="font-family: Symbol;">® (ù </span>A)) <span style="font-family: Symbol;">® ù </span>A)<o:p></o:p></li><li class="MsoNormal" style="">(<span style="font-family: Symbol;">ù </span>B <span style="font-family: Symbol;">Ù </span>A) <span style="font-family: Symbol;">Ù </span>B<o:p></o:p></li><li class="MsoNormal" style="">((A <span style="font-family: Symbol;">® (B ® </span>C)) <span style="font-family: Symbol;">® ((A ® </span>B) <span style="font-family: Symbol;">® (A ® </span>C)))<o:p></o:p></li><li class="MsoNormal" style="">((<span style="font-family: Symbol;">ù </span>A <span style="font-family: Symbol;">® </span>B) <span style="font-family: Symbol;">® (</span>B <span style="font-family: Symbol;">® A))</span><o:p></o:p></li><li class="MsoNormal" style="">((A <span style="font-family: Symbol;">Ù</span> B) <!--[if gte vml 1]><v:shape id="_x0000_i1049" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1049" width="23" border="0" height="16" /><!--[endif]-->A)<o:p></o:p></li></ol></ol> <p>7. Obtain the PCNF of the following<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr><!--[if gte vml 1]><v:shape id="_x0000_i1050" type="#_x0000_t75" alt="" style="'width:45.75pt;"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image016.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image210.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image016.gif" shapes="_x0000_i1050" width="61" align="absmiddle" border="0" height="25" /><!--[endif]--><o:p></o:p></nobr></li><li class="MsoNormal" style=""><!--[if gte vml 1]><v:shape id="_x0000_i1051" type="#_x0000_t75" alt="" style="'width:51.75pt;"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image017.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.2\Image\Image211.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image017.gif" shapes="_x0000_i1051" width="69" align="absmiddle" border="0" height="25" /><!--[endif]--><o:p></o:p></li><li class="MsoNormal" style=""><span style="font-family: Symbol;">ù</span> (A <!--[if gte vml 1]><v:shape id="_x0000_i1052" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1052" width="23" border="0" height="16" /><!--[endif]-->B)<o:p></o:p></li></ol></ol> <p>8. Obtain the product of sums canonical forms of the following.<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>(P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> R).<o:p></o:p></nobr></li><li class="MsoNormal" style="">(<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R <span style="font-family: Symbol;">Ù</span> S) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> S) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R<span style="font-family: Symbol;">Ù</span> S) <span style="font-family: Symbol;">Ú </span>
<br /> (P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> S).<o:p></o:p></li><li class="MsoNormal" style="">(<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (P <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> Q).<o:p></o:p></li><li class="MsoNormal" style="">(P <span style="font-family: Symbol;">Ù</span> Q) <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q <span style="font-family: Symbol;">Ù</span> R).<o:p></o:p></li></ol></ol> <p>9. Obtain the PDNF and PCNF of the following<o:p></o:p></p> <ol start="1" type="a"><ol start="1" type="a"><li class="MsoNormal" style=""><nobr>(Q <span style="font-family: Symbol;">Ù</span> (P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q)).<o:p></o:p></nobr></li><li class="MsoNormal" style="">P <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">®</span> (Q <span style="font-family: Symbol;">Ú</span> (<span style="font-family: Symbol;">ù</span> Q<span style="font-family: Symbol;">®</span> R))).<o:p></o:p></li><li class="MsoNormal" style="">(P<span style="font-family: Symbol;">®</span> (Q <span style="font-family: Symbol;">Ù</span> R)) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P<span style="font-family: Symbol;">®</span> (<span style="font-family: Symbol;">ù</span> Q <span style="font-family: Symbol;">Ù</span> <span style="font-family: Symbol;">ù</span> R)).<o:p></o:p></li><li class="MsoNormal" style="">P <span style="font-family: Symbol;">®</span> (P <span style="font-family: Symbol;">Ù</span> (Q<span style="font-family: Symbol;">®</span> P)).<o:p></o:p></li><li class="MsoNormal" style="">(Q<span style="font-family: Symbol;">®</span> P) <span style="font-family: Symbol;">Ù</span> (<span style="font-family: Symbol;">ù</span> P <span style="font-family: Symbol;">Ù</span> Q).<o:p></o:p></li></ol></ol> Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com2tag:blogger.com,1999:blog-7399621945608861143.post-34199954326824168012008-08-12T04:17:00.000-07:002008-12-23T01:55:02.556-08:00MATHEMATICAL LOGIC<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///D:%5CDOCUME%7E1%5Cshesu04%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C06%5Cclip_filelist.xml"><link rel="Edit-Time-Data" href="file:///D:%5CDOCUME%7E1%5Cshesu04%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C06%5Cclip_editdata.mso"><!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--><title>© Moreniche</title><o:smarttagtype namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="country-region"></o:smarttagtype><o:smarttagtype namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="City"></o:smarttagtype><o:smarttagtype namespaceuri="urn:schemas-microsoft-com:office:smarttags" name="place"></o:smarttagtype><!--[if gte mso 9]><xml> <o:documentproperties> <o:author>Marcus Polo</o:Author> <o:version>11.9999</o:Version> </o:DocumentProperties> </xml><![endif]--><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> <w:usefelayout/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><!--[if !mso]><object classid="clsid:38481807-CA0E-42D2-BF39-B33AF135CC4D" id="ieooui"></object> <style> st1\:*{behavior:url(#ieooui) } </style> <![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:SimSun; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:ËÎÌå; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:"\@SimSun"; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:"\@Arial Unicode MS"; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:SimSun;} p {mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:SimSun;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:60256761; mso-list-template-ids:1300818908;} @list l1 {mso-list-id:684408348; mso-list-template-ids:-854416412;} @list l2 {mso-list-id:872620921; mso-list-template-ids:430240742;} @list l2:level1 {mso-level-start-at:8; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l3 {mso-list-id:1421564920; mso-list-template-ids:400581896;} @list l4 {mso-list-id:1439183926; mso-list-template-ids:-1967244532;} @list l5 {mso-list-id:1712731443; mso-list-template-ids:1670913204;} @list l5:level1 {mso-level-start-at:7; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in;} @list l6 {mso-list-id:1997413616; mso-list-template-ids:-1465725484;} @list l7 {mso-list-id:2114082344; mso-list-template-ids:196365964;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> <p><a name="1.1.1_Introduction"><b>Introduction</b></a><b><o:p></o:p></b></p> <p style="line-height: 150%;">One of the main aims of logic is to provide rules by which one can determine whether any particular argument or reasoning is valid (correct). <o:p></o:p></p> <p style="line-height: 150%;">Logic is concerned with all kinds of reasonings, whether they are legal arguments or mathematical proofs or conclusions in a scientific theory based upon a set of hypothesis. Based on the diversity of their application, these rules are called rules of inference. In logic we are concerned with the forms of the arguments. The theory of inference is formulated in such a way that one should be able to decide about the validity of an argument by following the rules mechanically and independently of our own feelings about the argument. <o:p></o:p></p> <p style="line-height: 150%;">Any collection of rules or any theory needs a language. In this language these rules or theory can be explained. Natural languages are not used, and are not suitable for this purpose. Therefore, a formal language (object language) is developed and syntax is well defined in the object language. Symbols are used to define clearly in the object language. The object language requires the use of another language. We can use any of the natural languages like English to form the statements. Here English is called meta language. <o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p><a name="1.1.2____Atomic_Statements"><b>1.1.2 Atomic Statements</b></a> <o:p></o:p></p> <p style="line-height: 150%;"><i>The object language consists of a set of declarative sentences. These declarative sentences are called <b>primary or atomic or primitive statements</b></i>. These statements have only two truth vales TRUE (T or 1) and FALSE (F or 0). The atomic statements are denoted by the symbols A, B, C, D, E, …<o:p></o:p></p> <p style="line-height: 150%;"><i>The atomic statements are joined by using symbols, connectives and parenthesis. The connected statements are called <b>declarative sentences</b>.</i><o:p></o:p></p> <p style="line-height: 150%;"><i>If declarative sentences assume only one truth-value T or F, those sentences are called <b>statements</b></i>. Atomic statements are those which do not contain any connectives.<o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;">Consider the following examples :<o:p></o:p></p> <nobr> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">1.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><st1:place st="on"><st1:country-region st="on">India</st1:country-region></st1:place> is a country. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">2.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><st1:city st="on">London</st1:city> is the capital of <st1:place st="on"><st1:country-region st="on">Japan</st1:country-region></st1:place>. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">3.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->This statement is true. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">4.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->Open the gate. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">5.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><st1:place st="on"><st1:city st="on">Madurai</st1:city></st1:place> is an old city. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">6.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->Man will reach Mars by 2020. <o:p></o:p></p> </nobr> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;">Statements (1) and (2) have truth values true and false respectively. (3) is not a statement and one can not assign a definite truth value T or F. (4) is a command and not a statement. (5) is true in some part of the world and false in certain other parts. It is a statement. The truth-value of (6) could be determined only in the year 2020 or earlier if a man reaches Mars before that date.<o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;"><b>Proposition </b>: <i>A <b>proposition</b> is a statement that is either true or false but not both.</i><o:p></o:p></p> <p style="line-height: 150%;">Generally name is used as a name of the object.<o:p></o:p></p> <p style="line-height: 150%;">Consider an example :<o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">7.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->This chair is small. <o:p></o:p></p> <p style="line-height: 150%;">"This chair" is used as a name of the object.<o:p></o:p></p> <p style="line-height: 150%;">Consider another example :<o:p></o:p></p> <nobr> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">8.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->Srinivasan is a good man. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">9.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->"Srinivasan" contains ten letters. <o:p></o:p></p> </nobr> <p style="line-height: 150%;">The quality of a person is defined in (8) and the person’s name is Srinivasan. In (9), "Srinivasan" is used as a name of this name. (9) is about a name and not about a person.<o:p></o:p></p> <p style="line-height: 150%;"> <b> <o:p></o:p></b></p> <p style="line-height: 150%;"><a name="1.1.3____Connectives"><b>1.1.3 Connectives</b></a><b><o:p></o:p></b></p> <p style="line-height: 150%;">By using connectives, molecular or compound statements are formed from atomic statements. The statements are denoted by the capital letters A, B, C, D, E, …<o:p></o:p></p> <p style="line-height: 150%;"><b>Example 1:</b><o:p></o:p></p> <p style="line-height: 150%;">Let A be a proposition.<o:p></o:p></p> <p style="line-height: 150%;">A : Mr. Bill Clinton went to the White House.<o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.4_Negation"><b>1.1.4 Negation</b></a><b><o:p></o:p></b></p> <p style="line-height: 150%;">The symbol ‘<span style="font-family: Symbol;">ù</span> ’ is used to denote the negation. Alternate symbols used in the literature are ‘~’, a bar or "NOT", so that <span style="font-family: Symbol;">ù</span> P is written as ~P,<!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" spt="75" preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:formulas> <v:path extrusionok="f" gradientshapeok="t" connecttype="rect"> <o:lock ext="edit" aspectratio="t"> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" alt="" style="'width:10.5pt;"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image001.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image80.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image001.gif" shapes="_x0000_i1025" width="14" align="absmiddle" height="21" /><!--[endif]--> or NOT P.<o:p></o:p></p> <p style="line-height: 150%;"><b>Example 2:</b><o:p></o:p></p> <p style="line-height: 150%;">B : I went to my native place yesterday.<o:p></o:p></p> <p style="line-height: 150%;"><span style="font-family: Symbol;">ù</span> B: I did not go to my native place yesterday.<o:p></o:p></p> <p style="text-align: center; line-height: 150%;" align="center"><b>Truth table 1.1.4<o:p></o:p></b></p> <div align="center"> <table class="MsoNormalTable" style="width: 144.75pt;" width="193" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 44.25pt;" valign="top" width="59"> <p style="text-align: center; line-height: 150%;" align="center"><b>P</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 70.5pt;" valign="top" width="94"> <p style="line-height: 150%;"><b><span style="font-family: Symbol;">ù</span>P </b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 44.25pt;" valign="top" width="59"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 70.5pt;" valign="top" width="94"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 44.25pt;" valign="top" width="59"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 70.5pt;" valign="top" width="94"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.5_Conjunction"><b>1.1.5 Conjunction</b></a><o:p></o:p></p> <p style="line-height: 150%;"><i>The <b>conjunction of two statements</b> <b>A and B</b> is the statement <b>A </b></i><b><span style="font-family: Symbol;">Ù</span> <i>B </i></b><i>which is read as "A and B". The statement A </i><span style="font-family: Symbol;">Ù</span> <i>B has truth value T whenever both A and B have truth value T; otherwise it has the truth value F. </i><o:p></o:p></p> <p style="text-align: center; line-height: 150%;" align="center"><b>Truth table 1.1.5<o:p></o:p></b></p> <div align="center"> <table class="MsoNormalTable" style="width: 211.5pt;" width="282" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A </b><b><span style="font-family: Symbol;">Ù</span> B</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p style="line-height: 150%;">A : The University is conducting 25 Under Graduate courses.<o:p></o:p></p> <p style="line-height: 150%;">B : It has 15 Post Graduate courses.<o:p></o:p></p> <p style="line-height: 150%;"><b>Solution:</b><o:p></o:p></p> <p style="line-height: 150%;">The University is conducting 25 Under Graduate courses and it has 15<o:p></o:p></p> <p style="line-height: 150%;">Post Graduate courses.<b> </b><o:p></o:p></p> <p style="line-height: 150%;"><b> </b><o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.6_Disjunction"><b>1.1.6 Disjunction</b></a><o:p></o:p></p> <p style="line-height: 150%;"><i>Let A and B be propositions. The <b>disjunction of two statements</b> <b>A and B</b> is the statement <b>A </b></i><b><span style="font-family: Symbol;">Ú</span> <i>B </i></b><b><span style="font-family: Symbol;">Ú</span> <i>B </i></b><i>which is read as "A or B". The proposition <!--[if gte vml 1]><v:shape id="_x0000_i1026" type="#_x0000_t75" alt="" style="'width:31.5pt;height:12.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image002.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image82.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image002.gif" shapes="_x0000_i1026" width="42" align="absmiddle" height="17" /><!--[endif]-->has the truth value F only when both A and B have the truth value F; otherwise it is true. </i><o:p></o:p></p> <p style="text-align: center; line-height: 150%;" align="center"><b>Truth table 1.1.6<o:p></o:p></b></p> <div align="center"> <table class="MsoNormalTable" style="width: 211.5pt;" width="282" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A </b><b><span style="font-family: Symbol;">Ú</span> B</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 33%;" valign="top" width="33%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p style="line-height: 150%;">Consider an example :<o:p></o:p></p> <p style="line-height: 150%;">Baskaran shall play cricket or hockey tomorrow.<o:p></o:p></p> <p style="line-height: 150%;">The statement <!--[if gte vml 1]><v:shape id="_x0000_i1027" type="#_x0000_t75" alt="" style="'width:33pt;height:12.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image003.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image83.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image003.gif" shapes="_x0000_i1027" width="44" align="absmiddle" height="17" /><!--[endif]-->and <!--[if gte vml 1]><v:shape id="_x0000_i1028" type="#_x0000_t75" alt="" style="'width:33pt;height:12.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image004.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image84.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image004.gif" shapes="_x0000_i1028" width="44" align="absmiddle" height="17" /><!--[endif]-->have 2<sup>2</sup> possible combinations of truth values. If there are n distinct components in a statement, then the proposition has 2<sup>n</sup> possible combinations of truth values in order to obtain the truth table. A statement formula has no truth value. In the construction of formulas, the parenthesis will be used.<o:p></o:p></p> <p style="line-height: 150%;">For example,<o:p></o:p></p> <nobr> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">1.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><span style="font-family: Symbol;">ù</span> (<!--[if gte vml 1]><v:shape id="_x0000_i1029" type="#_x0000_t75" alt="" style="'width:31.5pt;height:12.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image005.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image85.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image005.gif" shapes="_x0000_i1029" width="42" align="absmiddle" height="17" /><!--[endif]--><sub> </sub>) means negation of <!--[if gte vml 1]><v:shape id="_x0000_i1030" type="#_x0000_t75" alt="" style="'width:31.5pt;height:12.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image005.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image85.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image005.gif" shapes="_x0000_i1030" width="42" align="absmiddle" height="17" /><!--[endif]-->. <o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">2.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><!--[if gte vml 1]><v:shape id="_x0000_i1031" type="#_x0000_t75" alt="" style="'width:132pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image006.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image86.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image006.gif" shapes="_x0000_i1031" width="176" align="absmiddle" height="25" /><!--[endif]-->means disjunction of <!--[if gte vml 1]><v:shape id="_x0000_i1032" type="#_x0000_t75" alt="" style="'width:61.5pt;height:16.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image007.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image87.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image007.gif" shapes="_x0000_i1032" width="82" align="absbottom" height="22" /><!--[endif]-->and<!--[if gte vml 1]><v:shape id="_x0000_i1033" type="#_x0000_t75" alt="" style="'width:63.75pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image008.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image88.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image008.gif" shapes="_x0000_i1033" width="85" align="absbottom" height="25" /><!--[endif]--> <o:p></o:p></p> <p style="line-height: 150%;"><!--[if gte vml 1]><v:shape id="_x0000_i1034" type="#_x0000_t75" alt="" style="'width:132pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image009.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image89.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image009.gif" shapes="_x0000_i1034" width="176" align="absmiddle" height="25" /><!--[endif]-->means conjunction of <!--[if gte vml 1]><v:shape id="_x0000_i1035" type="#_x0000_t75" alt="" style="'width:60pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image010.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image90.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image010.gif" shapes="_x0000_i1035" width="80" align="absbottom" height="25" /><!--[endif]-->and
<br /><!--[if gte vml 1]><v:shape id="_x0000_i1036" type="#_x0000_t75" alt="" style="'width:63.75pt;height:18.75pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image011.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image91.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image011.gif" shapes="_x0000_i1036" width="85" align="absmiddle" height="25" /><!--[endif]--><o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.7_Conditional_and_Bi-conditional"><b>1.1.7 Conditional and Bi-conditional</b></a><o:p></o:p></p> <p style="line-height: 150%;">Consider two statements A and B. <i>The statement <!--[if gte vml 1]><v:shape id="_x0000_i1037" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image92.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1037" width="50" align="absmiddle" height="18" /><!--[endif]-->is called <b>conditional statement</b> and is read as "If A, then B". The statement <!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image93.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1038" width="50" align="absmiddle" height="18" /><!--[endif]-->has truth value F when B has F and A has T; otherwise it has T.</i> A is called antecedent and B the consequent in <!--[if gte vml 1]><v:shape id="_x0000_i1039" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image93.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1039" width="50" align="absmiddle" height="18" /><!--[endif]-->.<o:p></o:p></p> <p style="text-align: center; line-height: 150%;" align="center"><b>Truth table 1.1.7(a)<o:p></o:p></b></p> <div align="center"> <table class="MsoNormalTable" style="width: 240pt;" width="320" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A </b><b><span style="font-family: Symbol;">®</span> B</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p style="line-height: 150%;"><!--[if gte vml 1]><v:shape id="_x0000_i1040" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image93.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1040" width="50" align="absmiddle" height="18" /><!--[endif]-->is represented by any one of the following :<o:p></o:p></p> <p style="margin-left: 1in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">1.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><nobr>B is necessary for A. <o:p></o:p></nobr></p> <p style="margin-left: 1in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">2.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->A is sufficient for B. <o:p></o:p></p> <p style="margin-left: 1in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">3.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->B if A. <o:p></o:p></p> <p style="margin-left: 1in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">4.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->A only if B. <o:p></o:p></p> <p style="margin-left: 1in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">5.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]-->A implies B. <o:p></o:p></p> </nobr> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;">Write the following statement in symbolic form.<o:p></o:p></p> <p style="line-height: 150%;"><b>Example 1:</b><o:p></o:p></p> <p style="line-height: 150%;">If Sindhu either reads the book or Vimal does the homework, then Viji will take the book to the college.<o:p></o:p></p> <p style="line-height: 150%;"><b>Solution:</b><o:p></o:p></p> <p style="line-height: 150%;">Denoting the statement as<o:p></o:p></p> <p style="line-height: 150%;">A : Sindhu reads the book.<o:p></o:p></p> <p style="line-height: 150%;">B : Vimal does the home work.<o:p></o:p></p> <p style="line-height: 150%;">C : Viji takes the book to the college.<o:p></o:p></p> <p style="line-height: 150%;">Symbolic form: <!--[if gte vml 1]><v:shape id="_x0000_i1041" type="#_x0000_t75" alt="" style="'width:64.5pt;height:16.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image013.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image94.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image013.gif" shapes="_x0000_i1041" width="86" align="absmiddle" height="22" /><!--[endif]--><o:p></o:p></p> <p style="line-height: 150%;"><b> <o:p></o:p></b></p> <p style="line-height: 150%;"><b>Example 2:<o:p></o:p></b></p> <p style="line-height: 150%;">If there is a fire, then the forest will be destroyed.<o:p></o:p></p> <p style="line-height: 150%;"><b>Solution:<o:p></o:p></b></p> <p style="line-height: 150%;">Denoting the statement as<o:p></o:p></p> <p style="line-height: 150%;">A : There is a fire.<o:p></o:p></p> <p style="line-height: 150%;">B : The forest will be destroyed.<o:p></o:p></p> <p style="line-height: 150%;">Symbolic form: <!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image93.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1042" width="50" align="absmiddle" height="18" /><!--[endif]--><o:p></o:p></p> <p style="line-height: 150%;"><i>The statement A<!--[if gte vml 1]><v:shape id="_x0000_i1043" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio13.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1043" width="23" border="0" height="16" /><!--[endif]--> B is called <b>bi-conditional statement</b>. It is translated as "A is necessary and sufficient for B". The statement A <!--[if gte vml 1]><v:shape id="_x0000_i1044" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio14.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1044" width="23" border="0" height="16" /><!--[endif]-->B has truth value T when both A and B has T or F. Otherwise it has truth value F.<o:p></o:p></i></p> <p style="text-align: center; line-height: 150%;" align="center"><b>Truth table 1.1.7(b)</b><o:p></o:p></p> <div align="center"> <table class="MsoNormalTable" style="width: 192pt;" width="256" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 28%;" valign="top" width="28%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 40%;" valign="top" width="40%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A <!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio15.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1045" width="23" border="0" height="16" /><!--[endif]-->B</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 28%;" valign="top" width="28%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 40%;" valign="top" width="40%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 28%;" valign="top" width="28%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 40%;" valign="top" width="40%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 28%;" valign="top" width="28%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 40%;" valign="top" width="40%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 28%;" valign="top" width="28%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 32%;" valign="top" width="32%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 40%;" valign="top" width="40%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p style="line-height: 150%;"><b>Example 3:</b><o:p></o:p></p> <p class="MsoNormal"><nobr>Construct truth table for (A<span style="font-family: Symbol;">®</span> B) <span style="font-family: Symbol;">Ù</span> (B<span style="font-family: Symbol;">®</span> A)</nobr><o:p></o:p></p> <div align="center"> <table class="MsoNormalTable" style="width: 359.25pt;" width="479" border="1" cellpadding="0" cellspacing="1"> <tbody><tr style=""> <td style="padding: 5.25pt; width: 12%;" valign="top" width="12%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 15%;" valign="top" width="15%"> <p style="text-align: center; line-height: 150%;" align="center"><b>B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 18%;" valign="top" width="18%"> <p style="text-align: center; line-height: 150%;" align="center"><b>A </b><b><span style="font-family: Symbol;">®</span> B</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 20%;" valign="top" width="20%"> <p style="text-align: center; line-height: 150%;" align="center"><b>B </b><b><span style="font-family: Symbol;">®</span> A</b><o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="line-height: 150%;"><b>(A</b><b><span style="font-family: Symbol;">®</span> B) </b><b><span style="font-family: Symbol;">Ù</span> (B</b><b><span style="font-family: Symbol;">®</span> A)</b><o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 12%;" valign="top" width="12%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 15%;" valign="top" width="15%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 18%;" valign="top" width="18%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 20%;" valign="top" width="20%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 12%;" valign="top" width="12%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 15%;" valign="top" width="15%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 18%;" valign="top" width="18%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 20%;" valign="top" width="20%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 12%;" valign="top" width="12%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 15%;" valign="top" width="15%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 18%;" valign="top" width="18%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 20%;" valign="top" width="20%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> </tr> <tr style=""> <td style="padding: 5.25pt; width: 12%;" valign="top" width="12%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 15%;" valign="top" width="15%"> <p style="text-align: center; line-height: 150%;" align="center">F<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 18%;" valign="top" width="18%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 20%;" valign="top" width="20%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> <td style="padding: 5.25pt; width: 36%;" valign="top" width="36%"> <p style="text-align: center; line-height: 150%;" align="center">T<o:p></o:p></p> </td> </tr> </tbody></table> </div> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.8_Well-formed_formulas_(wff)"><b>1.1.8 Well-formed formulas (wff)</b></a><o:p></o:p></p> <p style="line-height: 150%;"><i>A <b>well-formed formula</b> is nothing but a recursive definition of a statement formula. By applying the following rules a wff can be generated.</i><o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">1.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><i><nobr>A statement variable standing alone is a wff. </nobr></i><o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">2.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><i>If A is a wff, then </i><i><span style="font-family: Symbol;">ù</span> A is a wff. </i><o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">3.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><i>If A and B are wffs, then A</i><span style="font-family: Symbol;">Ù</span> <i>B, A</i><span style="font-family: Symbol;">Ú</span> <i>B, A</i><i><span style="font-family: Symbol;">®</span> B and A<!--[if gte vml 1]><v:shape id="_x0000_i1046" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio16.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1046" width="23" border="0" height="16" /><!--[endif]--> B are wffs. </i><o:p></o:p></p> <p style="margin-left: 0.5in; text-indent: -0.25in; line-height: 150%;"><!--[if !supportLists]--><span style=""><span style="">4.<span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><i>A string of symbols containing the statement variables, connectives and parenthesis is a wff. </i><o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p class="MsoNormal"><b><nobr>Example : </nobr></b><span style="font-family: Symbol;">ù</span> A<span style="font-family: Symbol;">Ú</span>B, A<span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> B, A<span style="font-family: Symbol;">®</span>(B<!--[if gte vml 1]><v:shape id="_x0000_i1047" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio11.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1047" width="23" border="0" height="16" /><!--[endif]--> C), A<span style="font-family: Symbol;">®</span> (B<span style="font-family: Symbol;">®</span>C)<o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.9_Tautologies"><b>1.1.9 Tautologies</b></a><o:p></o:p></p> <p style="line-height: 150%;"><i>A statement formula which is true regardless of the truth values of the statements which replace the variables in it is called a <b>universally valid formula or a tautology or a logical truth</b>.</i> <i>A statement formula which is false regardless of the truth values of the statements which replace the variables in it is called a <b>contradiction</b>. The negation of contradiction is a <b>tautology</b>.</i><o:p></o:p></p> <p style="line-height: 150%;">Consider two statement formulas, which are tautologies. If we assign any truth values to the variables of A and B, then the truth values of both A and B will be T. Therefore, the truth values of A <span style="font-family: Symbol;">Ù</span> B will be T, so that A <span style="font-family: Symbol;">Ù</span> B will be a tautology.<o:p></o:p></p> <p style="line-height: 150%;">A statement ‘A’ is said to tautologically imply a statement ‘B’ if and only if <!--[if gte vml 1]><v:shape id="_x0000_i1048" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image012.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image95.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image012.gif" shapes="_x0000_i1048" width="50" align="absmiddle" height="18" /><!--[endif]-->is a tautology. We shall denote this ideas by <!--[if gte vml 1]><v:shape id="_x0000_i1049" type="#_x0000_t75" alt="" style="'width:37.5pt;height:13.5pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image015.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Image96.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image015.gif" shapes="_x0000_i1049" width="50" align="absmiddle" height="18" /><!--[endif]-->which is read as "A implies B".<o:p></o:p></p> <p style="line-height: 150%;"> <o:p></o:p></p> <p style="line-height: 150%;"><a name="1.1.10_Equivalence_of_formulas"><b>1.1.10 Equivalence of Formulas</b></a><o:p></o:p></p> <ol start="1" type="1"><li class="MsoNormal" style=""><span style="font-family: Symbol;"><nobr>ù ù</nobr></span>P is equivalent to P.<o:p></o:p></li><li class="MsoNormal" style="">P <span style="font-family: Symbol;">Ú</span> P is equivalent to P.<o:p></o:p></li><li class="MsoNormal" style="">P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span>P is equivalent to Q <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span> Q.<o:p></o:p></li><li class="MsoNormal" style="">( P <span style="font-family: Symbol;">Ú</span> <span style="font-family: Symbol;">ù</span>P ) <span style="font-family: Symbol;">Ú</span> Q is equivalent to Q.<o:p></o:p></li></ol> <p style="line-height: 150%;">The equivalence of two formulas A and B is written as "A<span style="font-family: Symbol;">Û</span> B" which is read as "A is equivalent to B".<o:p></o:p></p> <p style="line-height: 150%;"><b>Examples:</b><o:p></o:p></p> <ol start="1" type="1"><li class="MsoNormal" style=""><nobr>P<span style="font-family: Symbol;">®</span> Q <span style="font-family: Symbol;">Û</span> <span style="font-family: Symbol;">ù</span>P <span style="font-family: Symbol;">Ú</span> Q.<o:p></o:p></nobr></li></ol> <p class="MsoNormal" style="margin-left: 0.5in;"> <o:p></o:p></p> <span style="font-size: 12pt; font-family: "Times New Roman";">P<!--[if gte vml 1]><v:shape id="_x0000_i1050" type="#_x0000_t75" alt="" style="'width:17.25pt;height:12pt'"> <v:imagedata src="file:///D:\DOCUME~1\shesu04\LOCALS~1\Temp\msohtml1\06\clip_image014.gif" href="file:///D:\Documents%20and%20Settings\shesu04\Desktop\dms\Sethuraman%20-%20MA109\Unit1\Section1.1\Image\Sectio12.gif"> </v:shape><![endif]--><!--[if !vml]--><img src="file:///D:/DOCUME%7E1/shesu04/LOCALS%7E1/Temp/msohtml1/06/clip_image014.gif" shapes="_x0000_i1050" width="23" border="0" height="16" /><!--[endif]--> Q </span><span style="font-size: 12pt; font-family: Symbol;">Û</span><span style="font-size: 12pt; font-family: "Times New Roman";"> (P</span><span style="font-size: 12pt; font-family: Symbol;">®</span><span style="font-size: 12pt; font-family: "Times New Roman";"> Q) </span><span style="font-size: 12pt; font-family: Symbol;">Ù</span><span style="font-size: 12pt; font-family: "Times New Roman";"> (Q</span><span style="font-size: 12pt; font-family: Symbol;">®</span><span style="font-size: 12pt; font-family: "Times New Roman";"> P)</span>Sumedhhttp://www.blogger.com/profile/11533458660230230361noreply@blogger.com1