Boolean algebra is one of the most interesting and important algebraic structure which has significant applications in switching

circuits, logic and many branches of computer science and engineering.

Boolean algebra can be viewed as one of the special type of lattice.

*Generally Boolean algebra is denoted by (B, *, Å , ',*

A complemented distributive lattice with 0 and 1 is called

A complemented distributive lattice with 0 and 1 is called

**Boolean algebra.**

**0**,

**1**).

**Example 1 :**

(

*P*(A), Ç , È , ', f, A) is a Boolean algebra. This is an important example of Boolean algebra [In fact the basic properties of

the (P (A), Ç , È , ' ) led to define the abstract concept of Boolean algebra]. Further, it can be proved that every finite Boolean

algebra must be isomorphic to (P (A), Ç , È , ' , f , A) for a suitably chosen finite set A. [refer [2]].

**Example 2:**

The structure ( B

^{n}= {0,1}

^{n}, *, Å ,

**1**,

**0**) is a Boolean algebra, where B

^{n}is an n-fold Cartesian product of

{0,1} and the operations *, Å , are defined below.

We have B

^{n}= {( l

_{1}, l

_{ 2}, … , l

_{ n}) / l

_{ r}= 0 or 1, 1 £ r £ n}

(i

_{ 1}, i

_{ 2}, … , i

_{n}) * (j

_{1 }, j

_{2}, … , j

_{n}) = (min (i

_{ 1}, j

_{1 }), min (i

_{ 2}, j

_{2}), .., min (i

_{n}, j

_{n}) )

(i

_{1}, i

_{2}, … , i

_{n}) Å (j

_{1}, j

_{2},…, j

_{n}) = (max ( i

_{1}, j

_{1}), max ( i

_{2}, j

_{2}), … , max ( i

_{ n}, j

_{n}) )

(l

_{ 1}, l

_{ 2}, l

_{ 3}, … , l

_{ n})' = (l

_{ 1}', l

_{ 2}', …. , l

_{ n}'),

**1**= (1,1, …, 1) is the greatest element of B

^{n}.

**0**= (0, 0,0, … ,0) is the least element of B

^{n}.

Since B is distributive, B

^{n}is distributive. From the definition of unary operation "

**'**",

it is clear that B

^{n}is complemented. Further, it has

**0**and

**1**. Thus, (B

^{n}, *, Å , ',

**0**,

**1**) is a Boolean algebra.

For the case n = 3 we have,

B

^{3}= {000, 100, 010, 001, 110, 101, 011, 111}.

The structure of the B

^{3}is given in the following Hasse diagram.

The Boolean algebra (B

^{n}, *, Å , ', 0,1) plays an important role in the construction of switching circuits, electronic circuits and

other applications. Also it can be proved that every finite Boolean algebra is isomorphic to the above Boolean algebra

(B

^{n}, *, Å , ',

**0**,

**1**), for some n. Thus, it is interesting to observe that number of elements in any finite Boolean algebra must be

always 2

^{n}, for some n.

*Let (B, *, Å , ' , 0,1) be a Boolean algebra and S Í B. If S contains the elements 0 and 1 and is closed under the*

operation *, Å and ' then (S, *, Å , ', 0,1) is called

operation *, Å and ' then (S, *, Å , ', 0,1) is called

**sub Boolean algebra**.

Example 1:

Example 1:

Consider the Boolean algebra (P ({1,2,3}), Ç , È , ' ,f , {1,2,3})

Then (S = {f , {1}, {2,3}, {1,2,3}}, Ç , È , ', f ,

**{1,2,3}) is also sub Boolean algebra.**

Similarly, S = ({f ,{3},{1,2},{1,2,3}}, Ç , È , ', f , {1,2,3}) is also sub Boolean algebra.

But (S = ({f, {1}, {2,3}, {1,2,3}}, Ç , È ,' , f , {1,2,3})) is not a sub Boolean algebra.

[Find why it is not a sub Boolean algebra].

*If we have two Boolean algebras (B*

by taking direct product of these two Boolean algebras. The

(B

_{1}, *_{1}, Å_{1}, ',**0**) and (B_{1}, 1_{1}_{2}, *_{2}, Å_{2}, ''**, 0**) then we can get new Boolean algebra_{2}, 1_{2}by taking direct product of these two Boolean algebras. The

**direct product**of these Boolean algebra is the Boolean algebra(B

_{1}´ B_{2}, *_{3}, Å_{3}, ''', 0_{3}, 1_{3}); where for any two elements (a_{1}, b_{1}) and (a_{2}, b_{2}) Î B_{1}´ B_{2},(a

_{1},b

_{1}) * (a

_{2},b

_{2}) = (a

_{1 }*

_{1 }a

_{2}, b

_{1 }*

_{2 }b

_{2})

(a

_{1},b

_{1})Å (a

_{2},b

_{2}) = (a

_{1 }Å

_{ 1 }a

_{2 }, b

_{1 }Å

_{ 2 }b

_{2})

(a

_{1},b

_{1}) ''' = (a

_{1}', b

_{1}'')

0

_{3}=(0

_{1},0

_{2})

**and**

**1**

_{3}=(1

_{1},1

_{2}).

*'*

Let (B, *, Å ,

Let (B, *, Å ,

*, 0*Ç

_{B}, 1_{B}) and (A,

*,*È

*, - , 0*

f : B A is called a

f (a * b) = f (a) Ç f (b).

_{A }, 1_{A}) be two Boolean algebra. A mappingf : B A is called a

**Boolean homomorphism**if f preserves all the Boolean operations, that is,f (a Å b) = f (a) È f (b).

f (0

_{B}) = 0

_{A.}

f (1

_{B}) = 1

_{A.}

A bijective Boolean homomorphism is called

A bijective Boolean homomorphism is called

**Boolean isomorphism.**

**Exercise:**

- Prove that mapping f defined in the exercise 1 of the Section 3.3.3 is a Boolean isomorphism.

- In every Boolean algebra, prove that (x * y)' = x' Å y' and (x Å y)' = x' * y', for all x, y.

- In a Boolean algebra B, for all x, y Î B, prove that

x £_{ }y_{ }x * y = 0_{}x' Å y = 1_{}x * y = x_{}x Å y = y.

fig ( i ) fig ( ii ) fig ( iii )

- Show that in a Boolean algebra the following are equivalent for any a and b.

( i ) a' Ú b = 1

( ii ) a b' = 0.

- The Boolean algebras (P (A
_{3}), Ç , È, ', A_{3}, f) and D_{30}, where A_{3}= {1,2,3} are (Boolean) isomorphic? Justify your

answer.