Q45 / 75 · JEE Main 2025 (28 Jan Shift 1) · Mathematics

The relation $R=\{(x, y): x, y \in \mathbb{Z}$ and $x+y$ is even $\}$ is:

  • A
    reflexive and symmetric but not transitive
  • Correct answer
    B
    an equivalence relation
  • C
    symmetric and transitive but not reflexive
  • D
    reflexive and transitive but not symmetric

Solution

For reflexive $(x, x) \in \mathbb{R}, x \in \mathbb{Z}$ $\Rightarrow x+x=2 x \rightarrow \text { even }$ For symmetric of $(x, y) \in \mathbb{R}$ then $(y, x) \in \mathbb{R}$ when $x, y \in \mathbb{Z}$ $x+y \rightarrow$ even $\Rightarrow y+x \rightarrow \text { even }$ for transitive if $(x, y) \in \mathbb{R} \Rightarrow x+y \rightarrow$ even $(y, z) \in \mathbb{R} \Rightarrow y+z \rightarrow \text { even }$ $x+2 y+z \rightarrow$ even $\Rightarrow x+z$ is even $\Rightarrow(x, z) \in \mathbb{R}$ $\Rightarrow \mathbb{R}$ is an equivalence relation.

Hence, the correct answer is option: 'B'.

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