Let the relation $R$ on $\mathbb{Z}$ be given by: $xRy$ if and only if $x - y \leq 0$. Is $R$ an equivalence relation?
Consider $x=-1$ and $y=5$. Then $x-y=(-1)-5=-6\leq0 \implies xRy$. However, $y - x = 5 - (-1)=6 > 0$, hence $y$ is not related to $x$ and $R$ is not symmetric. Conclude that $R$ is not an equivalence relation.
Is this correct?