JEE Mathematics 2025 Question with Solution

Given: The relation is

Find: Whether $R$ is reflexive, symmetric, and transitive.

To determine whether the relation is an equivalence relation, we check if it is reflexive, symmetric, and transitive.

Reflexive: For every integer $x$,

which is even. Hence, $(x, x) \in R$ for all $x \in \mathbb{Z}$. So the relation is reflexive.

Symmetric: If $(x, y) \in R$, then $x + y$ is even. Since

$y + x$ is also even. Therefore, $(y, x) \in R$. So the relation is symmetric.

Transitive: If $(x, y) \in R$ and $(y, z) \in R$, then $x + y$ and $y + z$ are even. Let

where $m, n \in \mathbb{Z}$. Subtracting,

so $x - z$ is even. Hence $x$ and $z$ have the same parity, therefore $x + z$ is even. Thus $(x, z) \in R$, so the relation is transitive.

Since the relation is reflexive, symmetric, and transitive, $R$ is an equivalence relation. Therefore, the correct option is D.

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

Find: The correct classification of the relation.

Step 1: Check reflexivity by seeing whether $x + x$ is even for all integers $x$.

Step 2: Check symmetry by seeing whether from $x + y$ even, it follows that $y + x$ is even.

Step 3: Check transitivity by verifying that if $x + y$ and $y + z$ are even, then $x + z$ is even.

All three properties hold, so the relation is an equivalence relation. Hence, the correct option is D.