JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Mathematical Reasoning *

Whenever $B$ is true the consequent $(\sim A)\vee B$ is true, and whenever $B$ is false the implication holds vacuously, so $B\Rightarrow ((\sim A)\vee B)$ is true for every assignment: it is a tautology. The equivalents are therefore exactly the options that are themselves tautologies. Now $B\Rightarrow (A\Rightarrow B)$ is always true, since a true $B$ makes $A\Rightarrow B$ true; $A\Rightarrow ((\sim A)\Rightarrow B)$ is always true, because a true $A$ makes $(\sim A)$ false and so $(\sim A)\Rightarrow B$ true; and $B\Rightarrow ((\sim A)\Rightarrow B)$ is always true for the same reason. In contrast $A\Rightarrow (A\iff B)$ fails when $A$ is true and $B$ is false, so it is not a tautology. The statements equivalent to the given one are $B\Rightarrow (A\Rightarrow B)$, $A\Rightarrow ((\sim A)\Rightarrow B)$ and $B\Rightarrow ((\sim A)\Rightarrow B)$.