JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Mathematical Reasoning *

Work from the inside out. The bracket $(p\wedge q)\vee (p\vee \sim q)$ simplifies because $(p\wedge q)\vee p=p$ by absorption, leaving $p\vee \sim q$. The whole expression is then $(p\vee \sim q)\wedge (\sim p\wedge \sim q)$. Distributing $(\sim p\wedge \sim q)$ over the bracket, the piece with $p$ gives $p\wedge \sim p\wedge \sim q$, which is false, while the piece with $\sim q$ gives $\sim q\wedge \sim p\wedge \sim q=\sim p\wedge \sim q$. The surviving term is $(\sim p)\wedge (\sim q)$.