If $p, q$ and $r$ are three propositions, then which of the following combination of truth values of $\mathrm{p}, \mathrm{q}$ and r makes the logical expression $\{(p \vee q) \wedge((\sim p) \vee r)\} \rightarrow((\sim q) \vee r)$ false ?

(A) $\mathrm{p}=\mathrm{T}, \mathrm{q}=\mathrm{F}, \mathrm{r}=\mathrm{T}$

(B) $\mathrm{p}=\mathrm{T}, \mathrm{q}=\mathrm{T}, \mathrm{r}=\mathrm{F}$

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Mathematical Reasoning *

p, q

and

r

are three propositions, then which of the following combination of truth values of

p, q

and r makes the logical expression

{(p \lor q) \land ((\sim p) \lor r)} \to ((\sim q) \lor r)

false ?

Answer: (C)

An implication is false only when its antecedent is true and its consequent false. The consequent

(\sim q) \lor r

is false exactly when

q

is true and

r

is false. With

q = T, r = F

, the antecedent

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

becomes

T \land (\sim p) =\sim p

, which is true only when

p

is false. The expression is therefore false precisely for

p = F, q = T, r = F

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Single correct option