Question

Explain what is Boolean algebra?

Answer 1

Definition of Boolean algebra

A non-empty set $B$ on which two binary operations $+$(addition), $.$(multiplication) and one unary operation $\text{'}$(complementation) are defined, is said to be a Boolean algebra if the following postulates are satisfied :

1. $+$and $.$ are commutative.

i.e. $a+b=b+aforalla,b\in B$

and, $a.b=b.aforalla,b\in B$

2. $+$is distributive over $.$

i.e. $a+(b.c)=(a+b).(a+c)foralla,b,c\in B$

3. $.$is distributive over $+$

i.e. $a.(b+c)=a.b+a.cforalla,b,c\in B$

4. There exist distinct elements $0$ and $I$ in $B$ such that,

$a+0=aforalla\in B$ and

$a.I=aforalla\in B$

5. The operation $\text{'}$ satisfies :

$a+a\text{'}=1foralla\in B$ and

$a.a\text{'}=0foralla\in B$

The Boolean algebra is denoted by $(B,+,.,\text{'})$