Question

If $R$ be a relation defined as $aRbiff|a-b|>0$, then the relation is

• A. reflexive
• B. symmetric
• C. transitive
• D. symmetric and transitive

Answer 1

Answer: (D)

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