Verified Solution • 2025–26 Session moderate

QJEEM23Ma636e: JEE Main 2023 Mathematics - Class 11 JEEM23Ma636e Solution: Among the two statements ( S 1 ) : ( p ⇒ q ) ∧ ( p ∧ ( ∼ q ) )…

Among the two statements

$(S 1) : (p \Rightarrow q) \land (p \land (\sim q))$ is a contradiction and

$(S 2) : (p \land q) \lor ((\sim p) \land q) \lor (p \land (\sim q)) \lor ((\sim p) \land (\sim q))$ is a tautology

Multiple Choice Options

A both are false.
B only (S1) is true.
C both are true.
D only (S2) is true.

Step-by-Step Verified Solution

\begin{aligned} S_{1} : & (p \Rightarrow q) \land (p \land \sim q) \\ \equiv (\sim p \lor q) \land (p \land \sim q) \\ \equiv (\sim p \land p \land \sim q) \lor (q \land p \land \sim q) \\ \equiv (f \land \sim q) \lor (f \land p) \\ \equiv f \lor f \equiv f \\ S_{2} : & (p \land q) \lor (\sim p \land q) \lor (p \land \sim q) \lor (\sim p \land \sim q) \\ \equiv ((p \lor \sim p) \land q) \lor ((p \lor \sim p) \land \sim q) \\ \equiv (t \land q) \lor (t \land \sim q) \equiv q \lor \sim q \equiv t \end{aligned}

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