JEE Main 2025 Jan 28 Shift 1, Mathematics Q7: Relations

Check all three properties on integers:

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

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

Transitive: If $x+y$ and $y+z$ are both even, then their sum $(x+y)+(y+z)=(x+z)+2y$ is even, so $x+z$ is even.

All three hold, so $R$ is an equivalence relation.