Let $\Delta, \nabla \in\{\wedge, \vee\}$ be such that $(\mathrm{p} \rightarrow \mathrm{q}) \Delta(\mathrm{p} \nabla \mathrm{q})$ is a tautology. Then

(C) $\Delta=\vee, \nabla=\vee$

(A) $\Delta=\wedge, \nabla=\vee$

(B) $\Delta=\vee, \nabla=\wedge$

(D) $\Delta=\wedge, \nabla=\wedge$

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Mathematical Reasoning *

Let

Δ, \nabla \in {\land, \lor}

be such that

(p \to q) Δ (p \nabla q)

is a tautology. Then

Answer: (C)

Recall that

p \to q \equiv \neg p \lor q

. Taking both connectives as

\lor

, the statement becomes

(\neg p \lor q) \lor (p \lor q) = (\neg p \lor p) \lor q

, which is true for every assignment and hence a tautology. Any choice using

\land

fails: with

p

true and

q

false,

p \to q

is false, and conjoining it makes the whole statement false. The tautology arises only when both

Δ

and

\nabla

are

\lor

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Single correct option