JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Sets Relations and Functions

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

Reflexive: $x+x=2x$ is always even, so $(x,x)\in R$ for every $x$.

Symmetric: if $x+y$ is even then $y+x=x+y$ is even too, so $(x,y)\in R\Rightarrow (y,x)\in R$.

Transitive: if $x+y$ is even and $y+z$ is even, then $x,y$ have the same parity and $y,z$ have the same parity, so $x,z$ have the same parity, making $x+z$ even; hence $(x,z)\in R$.

Since $R$ is reflexive, symmetric and transitive, it is an equivalence relation.