Let $\mathrm{p}, \mathrm{q}, \mathrm{r}$ be three logical statements. Consider the compound statements $S_{1}:((\sim p) \vee q) \vee((\sim p) \vee r)$ and $\mathrm{S}_{2}: \mathrm{p} \rightarrow(\mathrm{q} \vee \mathrm{r})$ Then, which of the following is NOT true ?

(C) If $\mathrm{S}_{2}$ is False, then $\mathrm{S}_{1}$ is True

(A) If $\mathrm{S}_{2}$ is True, then $\mathrm{S}_{1}$ is True

(B) If $\mathrm{S}_{2}$ is False, then $\mathrm{S}_{1}$ is False

(D) If $\mathrm{S}_{1}$ is False, then $\mathrm{S}_{2}$ is False

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Mathematical Reasoning *

Let

p, q, r

be three logical statements. Consider the compound statements

S_{1} : ((\sim p) \lor q) \lor ((\sim p) \lor r)

and

S_{2} : p \to (q \lor r)

Then, which of the following is NOT true ?

Answer: (C)

Simplify

S_{1} : ((\sim p) \lor q) \lor ((\sim p) \lor r) \equiv (\sim p) \lor (q \lor r)

.

By the conditional law,

S_{2} : p \to (q \lor r) \equiv (\sim p) \lor (q \lor r)

.

Thus

S_{1} \equiv S_{2}

. They are logically equivalent, so they have the same truth value in all cases.

Option (C) states "If

S_{2}

is False, then

S_{1}

is True", which contradicts their equivalence. This is NOT true.

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Single correct option