Question

Prove the following Boolean expression using the laws and theorems of Boolean algebra.

`("A" + "B") ("A" + overline"B")` = A

Answer 1

Expand using distributive law

`(A+B)(A+barB) = A(A+barB) + B(A+barB)`

`= (A A+AbarB) + (BA+BbarB)`

`(A+AbarB) + (BA+0)`

`= A+AbarB +AB`

From Absorption Law, A + AB = A

`A+AbarB + AB = A+A(barB+B)`

`barB + B=1`

A + A(1) = A + A = A

Hence proved:

`(A+A(1) = A+A=A`