GATE CS 2026 Set 2 — Question 26
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
Let be a binary relation on the set , where if the product of and is square of an integer. Which of the following properties is/are satisfied by ?
Correct answer
(A) Reflexive; (B) Symmetric; (C) Transitive
Solution
A binary relation on a set is defined as for some integer .
1.Reflexive: For any , , which is a perfect square. Thus, for all . is reflexive.
2.Symmetric: If , then . Since multiplication is commutative, , so . is symmetric.
3.Transitive: If and , then and . Multiplying gives . Since are positive integers, must be a rational number whose square is an integer, which implies is an integer. Thus is a perfect square. is transitive.
4.Antisymmetric: For , so . Also because . However, . Thus is not antisymmetric.
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