Boolean Algebra as an Algebraic Structure

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Boolean Algebra is an algebraic structure defined by a set of elements B, together with two operations, + and . satisfying the following axioms (Hunington postulates) .

Axiom 1:

(a) Closure with respect to +, that is if x, y B, x + y B.

(b) Closure with respect to, that is x, y B x . y B.

Axiom 2:

(a) There exists element 0B such that: x + 0 = 0 + x = x, x B.

(0 is the identity for +)

(b) There exists element 1 B such that: x . 1 = 1 . x = x.

(1 is the identity for ' . ')

For all x, y, z B.

Axiom 3:

(a) Commutative w.r.t + : x + y = y + x

(b) Commutative w.r.t . : x . y = y . x

Axiom 4:

(a) . is distributive over + : x . (y+z) = x . y + x . z

(b) + is distributive over . : x + (y . z) = (x+y) . (x+z)

Axiom 5:

" x B, $ x' B such that (a) x + x' = 1 and (b) x.x' = 0.

x' is called the complement or negation of x.

Axiom 6:

There exists at least two elements x and y such that x y.

Note:

To prove a set to be a Boolean algebra, we have to prove all the above six properties to be true.

Whenever we say B is a Boolean algebra, it should be understood that B is accompanied with two operations satisfying all the above six properties.

Two-valued Boolean Algebra

A two valued Boolean algebra is defined on a set of two elements B = {0, 1}, with rules for + and . as shown in the following tables.

These rules are exactly the same as the AND, OR and NOT operations.

Note:

• Closure is obvious from the tables, since the results of each operation is either 1 or 0.

• Also observe from the tables:

(a) 0 + 0 = 0, 0 + 1 = 1 + 0 = 1 0 is the identity w.r.t +.

(b) 1.1 = 1, 1.0 = 0.1 = 0 1 is the identity w.r.t .

• For 0, 1B, 0 + 1 = 1 + 0

0 . 1 = 1 . 0

Commutative law is satisfied for both + and .

• (a)

Since the last two columns are identical x.(y + z) = (x.y) + (x.z)

(b) Similarly it can be shown that

x + (y.z) = (x + y) . (x + z)

• Also from the complement table

x . x' = 0

(a) x + x^' = 1. since 0 + 0^' = 0 + 1 = 1, 1 + 1^' = 1 + 0 = 1

(b) x.x^' = 0, since 0.0^' = 0.1 = 0, 1.1^' = 1.0 = 0

Thus, we have a two valued Boolean algebra having a set of two elements 1 and 0.