Give an example of a relation which is reflexive and transitive but not symmetric.

(N/A) Define a relation $R$ on the set of real numbers $\mathbb{R}$ as:
$R = \{(a, b) : a^3 \geq b^3\}$
$1$. Reflexivity: For any $a \in \mathbb{R}$,$a^3 = a^3$,so $(a, a) \in R$. Thus,$R$ is reflexive.
$2$. Symmetry: Consider $(2, 1) \in R$ because $2^3 = 8 \geq 1^3 = 1$. However,$(1, 2) \notin R$ because $1^3 = 1 < 2^3 = 8$. Thus,$R$ is not symmetric.
$3$. Transitivity: Let $(a, b) \in R$ and $(b, c) \in R$.
This implies $a^3 \geq b^3$ and $b^3 \geq c^3$.
By the transitive property of inequality,$a^3 \geq c^3$.
Therefore,$(a, c) \in R$. Thus,$R$ is transitive.
Conclusion: The relation $R$ is reflexive and transitive but not symmetric.