##### Which of the following is true for the given Propositional function?

The question is:

The propositional function (~(P V Q) V (~P ^ Q) V P) is:       [NOTE: "^" means "AND" and "~" means "Negation".The problem should be solved without truth tables]

a)Tautology      b) Contradiction      c) Contingency      d)<=>P

I solved the question in below way:

STEP-1: I considered (~(P V Q) V (~P ^ Q)) as A and P as B.

STEP-2: I assumed the given function to be contrdiction and tried to prove (A V B) as false.

STEP-3:To make (A V B) to be false, make A and B both false.So,I considered P as false and substituted all the values of P in A as false.

STEP-4:Next,I assumed Q to be true in A.

STEP-5:When I finally solved by substituting T and F in A,I got A as T and B is F.Since (T V F) is T ,I assumed above propositional function  to be tautology.

My doubt starts here.

Similarly,following  the above mentioned  steps,if I substitute B as true,Iam getting (A V B) as false.(i.e  (~(P V Q) V (~P ^ Q)) as false).It should be contingency then.

But, the answer is given as A.

Akshay Saxena
20 Aug 2017 02:24 pm

Answer is A only if you are considering this by through contradiction Then your eqn A V B is true because you considered B as true no matter what value A has.
Simple soln to this answer :
Not(P or Q) OR (P or notP and P or Q) [P or not P = True]
Not(P or Q ) OR (P or Q)
True.

Surya Dhanraj
20 Aug 2017 03:06 pm

U CAN SOLVE IT LIKE THIS ALSO

(P+Q)'+(P'Q)+P

=P'Q'+P'Q+P

=P'(Q'+Q)+P

=P'+P

=1 IE TRUE

SO,IT IS TAUTOLOGY