Choose the correct choice(s) regarding the following propositional logic assertion S: S : ((P /\ Q) -> R) -> ((P /\ Q) -> (Q -> R)) A. S is neither a tautology nor a contradiction. B. S is a tautology. C. S is a contradiction. D. The antecedent of S is logically equivalent to the consequent of S.

GATE 2021 · Discrete Mathematics · Propositional Logic · medium

Answer: S is a tautology and its antecedent is logically equivalent to its consequent. Correct answers: B and D.

1. Simplify the antecedent A: A = (P /\ Q) -> R. By exportation: A == P -> (Q -> R)
2. Simplify the consequent B: B = (P /\ Q) -> (Q -> R). By exportation: B == P -> (Q -> (Q -> R)). Since Q -> (Q -> R) == Q -> R (by idempotent reasoning: Q -> (Q -> R) = ~Q \/ (~Q \/ R) = ~Q \/ R = Q -> R), we get B == P -> (Q -> R) == A.
3. Conclude about S: Since A == B, S = A -> B is equivalent to A -> A, which is a tautology. Also the antecedent is logically equivalent to the consequent.