Show that the relation $R$ in $R$ defined as $R =\{(a, b): a \leq b\},$ is reflexive and transitive but not symmetric.
Show that the relation $R$ in $R$ defined as $R =\{(a, b): a \leq b\},$ is reflexive and transitive but not symmetric.
Solution $4: R =\{( a , b ): a \leq b \}$
Clearly $(a, a) \in R$ $[$ as $a=a]$
$\therefore R$ is reflexive.
Now, $(2,4)\in R$ $($ as $2<4)$
But, $(4,2)\notin R$ as $4$ is greater than $2$.
$\therefore R$ is not symmetric. Now, let $(a, b),\,(b, c) \in R$
Then, $a \leq b$ and $b \leq c$
$\Rightarrow $ $a \leq c$
$\Rightarrow $ $(a, c) \in R$
$\therefore R$ is transitive.
Hence $R$ is reflexive and transitive but not symmetric
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