JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Mathematical Reasoning *

Using $P\to Q\equiv \sim P\vee Q$ twice, $A\to (B\to A)\equiv \sim A\vee (\sim B\vee A)\equiv (\sim A\vee A)\vee \sim B$, which is a tautology. The same reduction gives $A\to (A\vee B)\equiv \sim A\vee (A\vee B)\equiv (\sim A\vee A)\vee B$, again a tautology, so these two compound statements are equivalent. The other three are not tautologies: each of $A\to (A\leftrightarrow B)$, $A\to (A\to B)$ and $A\to (A\wedge B)$ is false when $A$ is true and $B$ is false. The statement equivalent to $A\to (B\to A)$ is $A\to (A\vee B)$.