Since, $R$ is defined as $aRb$ iff $\left|a-b\right|>0$.
Reflexive : $aRa$ iff $\left|a-a\right|>0$
Which is not true. So, $R$ is not reflexive.
Symmetric : $aRb$ iff $\left|a-b\right|>0$
Now, $bRa$ iff $\left|b-a\right|>0$
$\Rightarrow \left|a-b\right|>0 \Rightarrow aRb$
Thus, $R$ is symmetric.
Transitive : $aRb$ iff $\left|a-b\right|>0$
$bRc$ iff $\left|b-c\right|>0$
 
$aRb$ & $bRa$ $\Rightarrow aRa$ which is not correct, hence relation is not transitive.