The relation R in the set R of real numbers defined as : R =(a, b) :…

Given relation: $R = \{(a, b) : a \le b^2\}, a, b \in \mathbb{R}$

Check properties:

Reflexive: $a \le a^2$ for all $a \in \mathbb{R}$? - False, e.g., $a = 0.5 \Rightarrow 0.5 \le 0.25$ is false. - So, not reflexive.

Symmetric: If $a \le b^2$, does $b \le a^2$? - False, e.g., $a = 1, b = 2 \Rightarrow 1 \le 4$ but $2 \le 1$ is false. - So, not symmetric.

Transitive: If $a \le b^2$ and $b \le c^2$, does $a \le c^2$? - False, e.g., $a = 3, b = 2, c = 1 \Rightarrow 3 \le 4$, $2 \le 1$ false; counterexample shows transitivity fails. - So, not transitive.

Conclusion: neither reflexive, symmetric, nor transitive.