The number of choices of $\Delta \in\{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$, such that $(\mathrm{p} \Delta \mathrm{q}) \Rightarrow((\mathrm{p} \Delta \sim \mathrm{q}) \vee((\sim \mathrm{p}) \Delta \mathrm{q}))$ is a tautology, is

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Mathematical Reasoning *

The number of choices of

Δ \in {\land, \lor, \Rightarrow, \Leftrightarrow}

, such that

(p Δ q) \Rightarrow ((p Δ \sim q) \lor ((\sim p) Δ q))

is a tautology, is

Answer: (B)

For the implication to be a tautology,

(p Δ \sim q) \lor ((\sim p) Δ q)

must be true whenever

p Δ q

is true.

Testing each operator:

$Δ = \lor$ : $(p \lor \sim q) \lor ((\sim p) \lor q) = T \lor T = T$ ✓
$Δ =\Rightarrow$ : $(p \Rightarrow\sim q) \lor ((\sim p) \Rightarrow q)$ is always true ✓
$Δ = \land$ and $Δ =\Leftrightarrow$ fail.

Thus 2 choices work.

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Single correct option