We observe the following properties :
Reflexivity - Let a be an arbitrary element.
Then,
$\left|a-a\right|=0 > 0 \Rightarrow a R a$
This, R is not reflexive on R.
Symmetry - Let a and b be two distinct elements, then (a,b) $\in$R
$\Rightarrow \left|a-b\right|>0\Rightarrow \quad\left|b-a\right|>0$
$\quad\quad\quad \quad \quad \quad \left(\because \left|a-b\right|=\left|b-a\right|\right)$
$\Rightarrow \left(b,a\right)\in R$
Thus, $\left(a, b\right) \in R \Rightarrow \left(b, a\right) \in R.$ So, R is symmetric.
Transitivity - Let $\left(a, b\right) \in R$ and $\left(b, c\right) \in R.$
Then$\quad |a - b| > 0$ and $|b - c| > 0$
$\Rightarrow \quad\quad |a - c| > 0 \Rightarrow \left(a, c\right) \in R$
So, R is transitive.