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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.

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    Maybe you mean $e^{x-y}$ is an integer, maybe you mean $e^x-y$ is an integer. which is it?2012-12-30
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    I think it is about time you try to enhance seriously the way you write your posts: use LaTeX for mathematics (in the FAQ section you canfind some directions), use questions mark, periods where required, etc.2012-12-30
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    Based on the partial solution I guess that it is $e^{x-y}$.2012-12-30
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    Do you really think that to shwo reflexivity it suffices to consider merely the case $x=1$?2012-12-30
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    @HagenvonEitzen Thats my own understanding.What way would you have done it?2012-12-30
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    @Jack welch: you should prove it's true for any $x$ that $xRx$.2012-12-30

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