# Is $x - y \leq 0$ an equivalence relation on $\mathbb{Z}$?

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?

• You've done what's needed. It's correct. Move on to question $2$. Oct 13 at 22:09
• Worth noting that $x - y \leq 0 \iff x \leq y$. What gives this problem its flavor is that $R$ is both reflexive and transitive, but is not symmetric. Oct 13 at 23:57