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

(i) Reflexive

Let $a \in R,\;a \le a$ which is true. Therefore, $(a,\;a) \in R$

Thus, R is reflexive.

(ii) Symmetric

Let $a,\;b \in R\;\;\& \;\;(a,\;b) \in R$
Consider, $a \le b$ does not imply $b \le a$
$\Rightarrow$ $(a,\;b) \in R\;\;but\;\;(b,\;a)\not \in R$

Therefore, R is not symmetric.

(iii) Transitive

Let $a,\;b,\;c \in R$
If $(a,\;b) \in R \Rightarrow a \le b\;\;and\;\;(b,\;c) \in R \Rightarrow \;b \le c \Rightarrow a \le c$

$\Rightarrow$ $(a,\;c) \in R$

Thus, R is transitive.

Hence, R is reflexive and transitive but not symmetric.