Question

solve the logical expression

(a'b XOR ab) + a'b'c

reduce to IDNF expression

Answer 1

IDNF(Irredundant Disjunctive Normal Form)

If a logical expression consists of disjunction (OR operation) of one or more conjunction(AND) then the expression is said to be in Disjunctive Normal Form. In simple words, it is Sum of Products(SOP). that is one or more product terms are added together to get this form(it may include complemented or original form of a variable). For example, if A and B are two variables, AB , A'B, AB',A'B' are said to be the conjunctions and AB+A'B is a Disjunctive normal form

In the given question:

(a'b XOR ab) + a'b'c

= [(a'b)'ab+a'b(ab)' ]+a'b'c ( because, a XOR b = a'b +ab')

=(a''+b')ab+a'b(a'+b')+a'b'c (By Demorgan's law, (ab)' = a' + b')

=(a+b')ab+a'b(a'+b')+a'b'c (By involution law, (a')'=a )

=aab+abb'+a'ba'+a'bb'+a'b'c (Distributive law)

=ab+a'b+a'b'c ( By complement law,a . a' = 0, and By Idempotent law  a . a =a)

=ab+a'(b+b'c)

=ab+ '( (b+b') .(b+c))   (Distributive law)

=ab+a'(1.(b+c))   ( By complement law, b+b'=1)

=ab+a'b+a'c (Distributive law)

=(a+a')b+a'c  ( By complement law, a+a'=1)

=b+a'c