Question

Show that the relation $R$ on $R$ defined as $R=\left\{\right(a,b):a\le b\},$ is reflexive, and transitive but not symmetric.

Answer 1

$\text{Given}R=\left\{\right(a,b):a\le b\}$

For reflexive:
Clearly $R$ is reflexive as $a=a\forall a\in R.$

For transitive:
$\text{Let}(a,b)\in R\text{and}(b,c)\in R$
$\Rightarrow a\le b\cdots \left(1\right)\text{and}b\le c\cdots \left(2\right)$
$\text{From}\left(1\right)\text{and}\left(2\right),$ we get
$a\le c$
$\Rightarrow (a,c)\in R$
So, $R$ is transitive.

For symmetric:
$\text{Let}(a,b)\in R$
$\Rightarrow a\le b\forall a,b\in R$
This does not imply $b\le a\forall a,b\in R$
$\Rightarrow (b,a)\notin R$
So, $R$ is not symmetric.