The logical statement $[\sim(\sim p \vee q) \vee(p \wedge r) \wedge(\sim q \wedge r)]$ is equivalent to:

(A) $(p \wedge r) \wedge \sim q$

(B) $(\sim p \wedge \sim q) \wedge r$

(D) $(p \wedge \sim q) \vee r$

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Mathematical Reasoning *

The logical statement

[\sim (\sim p \lor q) \lor (p \land r) \land (\sim q \land r)]

is equivalent to:

Answer: (A)

Rewrite the inner negation with De Morgan:

\sim (\sim p \lor q) = p \land \sim q

. The statement becomes

[(p \land \sim q) \lor (p \land r)] \land (\sim q \land r)

. The bracket factors as

p \land (\sim q \lor r)

, and conjoining it with

(\sim q \land r)

-- which already makes

\sim q \lor r

true -- leaves

p \land \sim q \land r

. So the statement is equivalent to

(p \land r) \land \sim q

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option