JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Mathematical Reasoning *

We need to find $r$ such that $(p\wedge r)\iff (p\wedge (\sim q))\equiv (\sim p)$.

Testing $r=q$: construct the truth table for $(p\wedge q)\iff (p\wedge (\sim q))$.

When $p$ is T and $q$ is T: $(p\wedge q)=T$ and $(p\wedge (\sim q))=F$, so the biconditional is F.
When $p$ is T and $q$ is F: $(p\wedge q)=F$ and $(p\wedge (\sim q))=T$, so the biconditional is F.
When $p$ is F: both sides are F, so the biconditional is T.

This matches $(\sim p)$, confirming $r=q$.