JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Mathematical Reasoning *

An implication fails only when its antecedent is true and its consequent false. In the first compound statement the antecedent $\sim q\wedge (p\to q)$ can hold only when $q$ is false, and $p\to q$ with $q$ false forces $p$ to be false, so $\sim p$ is true whenever the antecedent is; the implication can never fail. In the second, $(p\vee q)\wedge \sim p$ can hold only when $p$ is false, and $p\vee q$ with $p$ false forces $q$ to be true, so the consequent $q$ holds whenever the antecedent does. Both compound statements are tautologies.