Question

Let $R$ be a relation defined as $aRb$ if $|a-b|>0.$ Then, the relation $R$ is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

Answer: (B)

$R$ is not reflexive since $|a-a|=0$ and so $|a-a|\ngtr 0$.
Thus $a/Ra$ for any real number $a$.
$R$ is symmetric since if $|a-b|>0$, then
$|b-a|=|a-b|>0$
Thus $aRb\Rightarrow bRa$
$R$ is not transitive. For example, consider the numbers $3,7,3$.
Then we have $3R7$ since $|3-7|=4>0$ and $7R3$ since $|7-3|=4>0$
But $3R3$ since $|3-3|=0$ so that $|3-3|\ngtr 0$.