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Q.
Let $R$ be a relation defined as $a R b$ if $|a-b|>0 .$ Then, the relation $R$ is
Relations and Functions - Part 2
Solution:
$R$ is not reflexive since $|a-a|=0$ and so $|a-a| \ngtr 0$.
Thus $a \,\not R \,a$ for any real number $a$.
$R$ is symmetric since if $|a-b|>0$, then
$|b-a|=|a-b|>0$
Thus $a \,R \,b \Rightarrow b\, R\, a$
$R$ is not transitive. For example, consider the numbers $3,7,3$.
Then we have $3\,R\, 7$ since $|3-7|=4>0$ and $7 \,R\, 3$ since $|7-3|=4>0$
But $3 \,\not R \,3$ since $|3-3|=0$ so that $|3-3| \ngtr 0$.