Let $R$ be a relation on the set $\Bbb R$ of real numbers where real numbers $x,y$ satisfy $xRy$ if and only if $e^{x-y}$ is an integer. Is $R$ an equivalence relation on $\Bbb R$? Is it a partial order?
I have proved it's reflexive let $x=1$. for reflexivity $xRx$.
so $e^{x-x}= e^{1-1}=e^0 =1$. This also satisfies the result being an integer. Therefore the relation is reflexive.
To check is the relation is symmetric it means. $xRy$ then $yRx$. The problem I am facing is I can't seem to find an $x$ and $y$ that satisfies $e^{x-y}$ being an integer.