Question

The following relations are defined on the set of real numbers check them for R,S,T.

aRb iff $|a-b|>0$

Answer 1

Given $aRb\text{iff}|a-b|>0;a,b\in R$
Reflexive :
We know $|a-a|\ngtr 0$
Hence, $a(\sim R)a$
Hence, R is not reflexive.
Symmetric:
Let $aRb$
$\Rightarrow |a-b|>0$
$\Rightarrow |b-a|>0$
$\Rightarrow bRa$
Hence, R is symmetric.
Transitive:
Let $aRb,bRc$
$\Rightarrow |a-b|>0;|b-c|>0$
$\Rightarrow |a-c|>0$
R is transitive.