JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Mathematical Reasoning *

A tautology is a statement that is always true. We check each option by constructing truth tables. For option (D): $q\Rightarrow ((\sim p)\vee q)$. When $q$ is true, $(\sim p)\vee q$ is always true (regardless of $p$), so the implication is true. When $q$ is false, the antecedent is false, making the implication vacuously true. Thus, option (D) is always true and is a tautology. Options (A), (B), and (C) can be shown to have at least one false entry in their truth tables.