The converse of the statement $((\sim p) \wedge q) \Rightarrow r$ is

(D) $(\mathrm{p} \vee(\sim \mathrm{q})) \Rightarrow(\sim \mathrm{r})$

(A) $(\sim \mathrm{r}) \Rightarrow \mathrm{p} \wedge \mathrm{q}$

(B) $(\sim \mathrm{r}) \Rightarrow((\sim \mathrm{p}) \wedge \mathrm{q})$

(C) $((\sim \mathrm{p}) \vee \mathrm{q}) \Rightarrow \mathrm{r}$

Mathematical Reasoning * — JEE Main · 2023 · 11 Apr 2023 · Shift 2 Question with Solution

The converse of the statement

((\sim p) \land q) \Rightarrow r

Answer: (D)

The converse of

A \Rightarrow B

B \Rightarrow A

, so the converse of

((\sim p) \land q) \Rightarrow r

r \Rightarrow ((\sim p) \land q)

. Expressing this implication as a disjunction,

r \Rightarrow ((\sim p) \land q) \equiv (\sim r) \lor ((\sim p) \land q)

. The same disjunction can be read as an implication with

(\sim r)

as consequent:

(\sim r) \lor ((\sim p) \land q) \equiv (\sim ((\sim p) \land q)) \Rightarrow (\sim r)

. By De Morgan's law

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

, so the converse is

(p \lor (\sim q)) \Rightarrow (\sim r)

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Single correct option