JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Mathematical Reasoning *

Every implication here rewrites as a disjunction: $\sim q\Rightarrow p$ is $q\vee p$, $\sim p\Rightarrow q$ is $p\vee q$, $q\Rightarrow p$ is $\sim q\vee p$, $p\Rightarrow \sim q$ is $\sim p\vee \sim q$ and $p\Rightarrow q$ is $\sim p\vee q$. Three of the four expressions then contain a complementary pair: $(\sim q\vee p)\vee (q\vee p)$ and $(\sim p\vee \sim q)\vee (q\vee p)$ both contain $q\vee \sim q$, and $(\sim p\vee q)\vee (q\vee p)$ contains $p\vee \sim p$, so each is true for every assignment. The remaining expression collapses to $(p\vee q)\vee (q\vee p)=p\vee q$, which is false when $p$ and $q$ are both false. The expression that is not a tautology is $(\sim p\Rightarrow q)\vee (\sim q\Rightarrow p)$.