Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) R if and only if p - s = q - r. Which one of the following is true about R? A) Both reflexive and symmetric B) Reflexive but not symmetric C) Not reflexive but symmetric D) Neither reflexive nor symmetric

Official answer: C) Not reflexive but symmetric. Explanation: Given the condition for ((p, q), (r, s)) R: p - s = q - r p + r = q + s 1. Reflexivity: For R to be reflexive, ((p, q), (p, q)) R must hold for all pairs of positive integers (p, q). This requires: p + p = q + q 2p = 2q p = q This is not true for arbitrary positive integers (e.g., for (1, 2), 1 + 1 2 + 2). Thus, R is not reflexive. 2. Symmetry: Assume ((p, q), (r, s)) R, which means p + r = q + s. For symmetry, ((r, s), (p, q)) R must hold, which requires: r + p = s + q Since addition of integers is commutative, p + r = q + s implies r + p = s + q. Thus, R is symmetric. Therefore, R is Not reflexive but symmetric, making C the correct option.

A) Both reflexive and symmetric

D) Neither reflexive nor symmetric

Engineering Mathematics · Sets, Relations, Lattices and Groups

Official IIT answer key · IIT Kanpur · Audited Aug 2026

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GATE CSE 2015 Set 3 Q41 · MCQ · 2 marks

Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true about $R$ ?

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Sets, Relations, Lattices and Groups

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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 3 Q41 · MCQ · 2 marks

Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true about $R$ ?

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

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

Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true about $R$ ?

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Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true about $R$ ?

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Let RRR be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s))∈R((p,q),(r,s)) \in R((p,q),(r,s))∈R if and only if p−s=q−rp - s = q - rp−s=q−r. Which one of the following is true about RRR?

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Let RRR be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s))∈R((p,q),(r,s)) \in R((p,q),(r,s))∈R if and only if p−s=q−rp - s = q - rp−s=q−r. Which one of the following is true about RRR?

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Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true about $R$ ?

Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true about $R$ ?