The number of values of \(\mathrm{r} \in\{\mathrm{p}, \mathrm{q}, \sim \mathrm{p}, \sim \mathrm{q}\}\) for which \(((\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \vee \mathrm{q})) \wedge((\mathrm{p} \wedge \mathrm{r}) \Rightarrow \mathrm{q})\) is a tautology, is:

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Mathematical Reasoning *

The number of values of

r \in {p, q, \sim p, \sim q}

for which

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

is a tautology, is:

Answer: (B)

The part

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

is always true because

q

sits in the consequent, so the statement is a tautology exactly when

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

is. For

r = q

this is

\sim p \lor \sim q \lor q \equiv T

, and for

r =\sim p

it is

\sim p \lor p \lor q \equiv T

; but

r = p

reduces it to

p \Rightarrow q

and

r =\sim q

(p \land \sim q) \Rightarrow q

, neither a tautology. Exactly

2

values of

r

make it a tautology.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Single correct option