Show that the relation $R$ in the set of real numbers $\mathbb{R}$ defined as $R = \{(a, b) : a \leq b\}$ is reflexive and transitive but not symmetric.

(N/A) Given the relation $R = \{(a, b) : a \leq b\}$ on the set of real numbers $\mathbb{R}$.
$1$. Reflexive: For any $a \in \mathbb{R}$,we know that $a \leq a$ is always true. Therefore,$(a, a) \in R$ for all $a \in \mathbb{R}$. Hence,$R$ is reflexive.
$2$. Symmetric: Consider $(2, 4) \in R$ because $2 \leq 4$. However,$(4, 2) \notin R$ because $4 \not\leq 2$. Since $(a, b) \in R$ does not imply $(b, a) \in R$,the relation is not symmetric.
$3$. Transitive: Let $(a, b) \in R$ and $(b, c) \in R$. This means $a \leq b$ and $b \leq c$. By the transitive property of inequality,$a \leq c$. Therefore,$(a, c) \in R$. Hence,$R$ is transitive.
Conclusion: The relation $R$ is reflexive and transitive but not symmetric.