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Discrete Mathematics

GATE CSE 2017 Set 1 · Q29

MCQ2 marksMedium

Time

Let pp, qq, and rr be propositions and the expression (p→q)→r(p \to q) \to r be a contradiction. Then, the expression (r→p)→q(r \to p) \to q is

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Official answer: D) always TRUE when q is TRUE.

Discrete Mathematics

If the expression (p→q)→r(p \to q) \to r is a contradiction (always FALSE), then for the conditional to be FALSE:

  1. The antecedent (p→q)(p \to q) must be TRUE.

  2. The consequent rr must be FALSE.

Now consider the expression (r→p)→q(r \to p) \to q:

  • Since rr is FALSE, the implication (r→p)(r \to p) is vacuously TRUE regardless of the truth value of pp.

  • Substituting (r→p)=TRUE(r \to p) = \text{TRUE} into the expression gives:

TRUE→q≡q\text{TRUE} \to q \equiv q

Thus, the expression (r→p)→q(r \to p) \to q has the same truth value as qq. Therefore, it is always TRUE when qq is TRUE.

Hence, the correct option is (D).

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This Discrete Mathematics previous year question is Q29 of the GATE CSE 2017 Set 1 paper. Find more GATE CSE PYQs on Discrete Mathematics in Engineering Mathematics, or browse all GATE CSE PYQs.

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Discrete Mathematics in GATE CSE

Discrete Mathematics has 50 GATE CSE PYQs across 23 papers from 2011 to 2026. 11 of them are 2-mark MCQs like this one, and GATE CSE 2017 Set 1 has 4.

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GateAI solution cardGATE CSE 2017 Set 1 · Q29 · MCQ

Let p, q, and r be propositions and the expression (p → q) → r be a contradiction. Then, the expression (r → p) → q is

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.