The compound statement $(\sim(\mathrm{P} \wedge \mathrm{Q})) \vee((\sim \mathrm{P}) \wedge \mathrm{Q}) \Rightarrow((\sim \mathrm{P}) \wedge(\sim \mathrm{Q}))$ is equivalent to

(A) $((\sim \mathrm{P}) \vee \mathrm{Q}) \wedge((\sim \mathrm{Q}) \vee \mathrm{P})$

(B) $(\sim \mathrm{Q}) \vee \mathrm{P}$

(C) $((\sim \mathrm{P}) \vee \mathrm{Q}) \wedge(\sim \mathrm{Q})$

(D) $(\sim \mathrm{P}) \vee \mathrm{Q}$

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Mathematical Reasoning *

The compound statement

(\sim (P \land Q)) \lor ((\sim P) \land Q) \Rightarrow ((\sim P) \land (\sim Q))

is equivalent to

Answer: (A)

Evaluating the antecedent

r = (\sim (P \land Q)) \lor ((\sim P) \land Q)

and the consequent

s = (\sim P) \land (\sim Q)

, the implication

r \Rightarrow s

is true only when

P

and

Q

hold the same truth value, giving the pattern

T, F, F, T

. The statement

((\sim P) \lor Q) \land ((\sim Q) \lor P)

expresses

(P \Rightarrow Q) \land (Q \Rightarrow P)

, true exactly when

P

and

Q

agree, with the same pattern

T, F, F, T

. The compound statement is therefore equivalent to

((\sim P) \lor Q) \land ((\sim Q) \lor P) .

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option