If the proposition p => q is true, then the truth value of the proposition ~p v (p => ~q), where ~ is negation, v is inclusive OR and => is implication, is A. True B. Multiple Values C. False D. Cannot be determined

GATE 1995 · Discrete Mathematics · Propositional Logic · medium

Answer: Cannot be determined — the formula is True when p is False but False when p=True and q=True

1. Identify valid assignments (p => q = True): Valid rows: (p=F,q=F), (p=F,q=T), (p=T,q=T). The row (p=T,q=F) is excluded.
2. Evaluate ~p v (p => ~q) for each valid row: (p=F,q=F): ~p=T => result T. (p=F,q=T): ~p=T => result T. (p=T,q=T): ~p=F, p=>~q=T=>F=F, so F v F = F.