Among the two statements (S1): \((p \Rightarrow q) \wedge (q \wedge (\sim q))\) is a contradiction and (S2): \((p \wedge q) \vee ((\sim p) \wedge q) \vee (p \wedge (\sim q)) \vee ((\sim p) \wedge (\sim q))\) is a tautology

Mathematical Reasoning * — JEE Main · 2023 · 12 Apr 2023 · Shift 1 Question with Solution

Among the two statements
(S1):

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

is a contradiction and
(S2):

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

is a tautology

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

Answer: (D)

Solution

In S1 the part

q \land (\sim q)

is false for every truth value of

q

, so

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

is false in every case; thus S1 is a contradiction and the statement S1 is true. In S2 the four disjuncts

(p \land q)

((\sim p) \land q)

(p \land (\sim q))

and

((\sim p) \land (\sim q))

cover all four combinations of truth values of

p

and

q

, so exactly one of them holds in each case and the disjunction is always true; thus S2 is a tautology and the statement S2 is true. Both statements are true.

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Single correct option

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Mathematical Reasoning *

Solution

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