Question

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

Answer 1

Given, $R=\left\{\right(a,b);a\le b\}$
Clearly $(a,a)\in R$ as $a=a$
$\therefore R$ is reflexive.

Now, $(2,4)\in R$ $($ As $2<4)$
But, $(4,2)\notin R$ because $4$ is greater than $2$
$\therefore R$ is not symmetric.
Now, let $(a,b),(b,c)\in R$
Then, $a\le b$ and $b\le c$
$\Rightarrow a\le c$
$\Rightarrow (a,c)\in R$
$\therefore R$ is transitive.
Hence, $R$ is reflexive and transitive but not symmetric.