Given the formula A B V A C B obtain equivalent formulas

Given the formula (( ~A -> B) V ((A & ~C) <-> B) ), obtain equivalent formulas in DNF and CNF.

Solution

(( ~A-> B)v((A &~C)<->B))

elliminating -> and <-> in above equation

by using a->b=~avb and a<->b =(a & b ) v ( ~a & ~b)

= (( ~A-> B)v((A &~C)<->B))

=((A v B ) v (( A & ~C & B ) v (~A v C & ~B ) )

lets simlify this

= ( A v B ) v ( A & ~C & B v ~A v C& ~B )

=  ( A v B ) v ( ~A v A & ~C & B v C& ~B )

=   A v B v ~A v (A & ~C & B) v( C& ~B )

= 1 v (A & ~C & B) v( C& ~B )=1