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JEE Main 202528 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
28 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Sets and Relations

JEE Main 2025 · 28 January · Morning Shift · Mathematics · Sets and Relations

The relation R=\ (x, y): x, y ∈ ℤ and x+y is even is:

The relation R={(x,y):x,y∈ZandR=\{(x, y): x, y \in \mathbb{Z} andx+yisevenis even is:

Options

  1. Areflexive and symmetric but not transitive
  2. Ban equivalence relation
  3. Csymmetric and transitive but not reflexive
  4. Dreflexive and transitive but not symmetric

Correct answer

B. an equivalence relation

Step-by-step solution

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