Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true? A) R is symmetric and reflexive but not transitive B) R is reflexive but not symmetric and not transitive C) R is transitive but not reflexive and not symmetric D) R is symmetric but not reflexive and not transitive

Official answer: D) R is symmetric but not reflexive and not transitive. Explanation: Given the definition $aRb a e b and (a, b) > 1 for positive integers a, b$: 1. Reflexivity: For any positive integer a, a and a are not distinct (a = a), so (a, a) R. Hence, R is not reflexive (it is irreflexive). 2. Symmetry: If aRb, then $a e b and (a, b) > 1. Since b e a and (b, a) = (a, b) > 1, it follows that bRa. Hence, R$ is symmetric. 3. Transitivity: Consider a = 2, b = 6, c = 3: - $a e b and (2, 6) = 2 > 1 2R6$. - $b e c and (6, 3) = 3 > 1 6R3$. - However, (2, 3) = 1, so (2, 3) R. Therefore, R is not transitive. Thus, R is symmetric but not reflexive and not transitive (Option D).

A) R is symmetric and reflexive but not transitive

B) R is reflexive but not symmetric and not transitive

C) R is transitive but not reflexive and not symmetric

D) R is symmetric but not reflexive and not transitive

Engineering Mathematics · Sets, Relations, Lattices and Groups

Official IIT answer key · IIT Kanpur · Audited Aug 2026

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GATE CSE 2015 Set 2 Q16 · MCQ · 1 mark

Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?

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Key concept

Sets, Relations, Lattices and Groups

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How GATE tests Sets, Relations, Lattices and Groups

In GATE CS, questions on sets, relations, and functions test formal mathematical rigor through algebraic verification of binary relation properties (reflexivity, symmetry, transitivity), set-theoretic properties of direct and inverse images of functions, and state-reachability or partition analysis on integer domain functions. The questions range from conceptual 1-mark MCQs on standard definitions and counterexamples to 2-mark algebraic or NAT problem-solving items involving equivalence classes and modular cycles.

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Engineering Mathematics · Sets, Relations, Lattices and Groups

Official IIT answer key · IIT Kanpur · Audited Aug 2026

Take topic test

GATE CSE 2015 Set 2 Q16 · MCQ · 1 mark

Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?

Select an option to check

Key concept

Sets, Relations, Lattices and Groups

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Sets, Relations, Lattices and Groups

Full Sets, Relations, Lattices and Groups guide →

More on Sets, Relations, Lattices and Groups

All 11 questions on Sets, Relations, Lattices and Groups →

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

Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?

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Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?

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Let RRR be the relation on the set of positive integers such that aRbaRbaRb if and only if aaa and bbb are distinct and have a common divisor other than 1. Which one of the following statements about RRR is true?

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Let RRR be the relation on the set of positive integers such that aRbaRbaRb if and only if aaa and bbb are distinct and have a common divisor other than 1. Which one of the following statements about RRR is true?

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Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?

Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?