Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

$R=\{(a, b) ; a \leq b\}$

Clearly $(a, a) \in R$ as $a=a$.

∴R is reflexive.

Now,

$(2,4) \in R(\operatorname{as} 2<4)$

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.