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GATE 2026 CS (CS2) – Question 26

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 1 mark · Multiple select

Let $R$ be a binary relation on the set $\{1,2,\dots,10\}$, where $(x,y) \in R$ if the product $xy$ is the square of an integer. Which of the following properties is/are satisfied by $R$?

  1. Reflexive
  2. Symmetric
  3. Transitive
  4. Antisymmetric

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Show answer and explanation

Correct answer: (A) Reflexive; (B) Symmetric; (C) Transitive

Explanation

The relation is reflexive because for every $x$, the product $x\cdot x=x^2$ is a perfect square. It is symmetric because if $xy$ is a square, then $yx$ is also a square. It is transitive because if $xy$ and $yz$ are both perfect squares, then $(xy)(yz)=y^2xz$ is a perfect square; dividing by the square $y^2$ shows $xz$ is also a perfect square. It is not antisymmetric because, for example, $(2,8)\in R$ and $(8,2)\in R$ but $2\ne 8$. Therefore, (A), (B), and (C) are true.