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?

  • 9
    $\begingroup$ You've done what's needed. It's correct. Move on to question $2$. $\endgroup$ Oct 13 at 22:09
  • 2
    $\begingroup$ 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. $\endgroup$ Oct 13 at 23:57

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