Statement $(P \Rightarrow Q) \wedge(R \Rightarrow Q)$ is logically equivalent to

(A) $(P \vee R) \Rightarrow Q$

(B) $(P \Rightarrow R) \wedge(Q \Rightarrow R)$

(C) $(P \Rightarrow R) \vee(Q \Rightarrow R)$

(D) $(P \wedge R) \Rightarrow Q$

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Mathematical Reasoning *

Statement

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

is logically equivalent to

Answer: (A)

Using

P \Rightarrow Q \equiv \sim P \lor Q

, the statement becomes

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

. Factoring

Q

by the distributive law gives

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

, and by De Morgan's law

\sim P \land \sim R = \sim (P \lor R)

, so the expression equals

\sim (P \lor R) \lor Q

, which is

(P \lor R) \Rightarrow Q

. Hence the given statement is logically equivalent to

(P \lor R) \Rightarrow Q

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Single correct option