JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Mathematical Reasoning *

We check each option. For option (A) with $\ast =\vee ,\odot =\wedge$: $(p\vee q)\wedge (p\vee \sim q)\equiv p\vee (q\wedge \sim q)\equiv p$, which is not a tautology. For option (B) with $\ast =\vee ,\odot =\vee$: $(p\vee q)\vee (p\vee \sim q)$. This is always true because for any truth values of $p$ and $q$, at least one of $q$ or $\sim q$ is true, making the entire expression true. This is a tautology. Therefore, the answer is $(\vee ,\vee )$.