Check whether the relation $R$ in $\mathbb{R}$ defined by $R = \{(a, b) : a \leq b^3\}$ is reflexive,symmetric,or transitive.

(D) The relation is defined as $R = \{(a, b) : a \leq b^3\}$.
$1$. Reflexivity: $A$ relation $R$ is reflexive if $(a, a) \in R$ for all $a \in \mathbb{R}$.
Consider $a = \frac{1}{2}$. Since $\frac{1}{2} > (\frac{1}{2})^3 = \frac{1}{8}$,the condition $a \leq a^3$ is not satisfied.
Thus,$(\frac{1}{2}, \frac{1}{2}) \notin R$. Therefore,$R$ is not reflexive.
$2$. Symmetry: $A$ relation $R$ is symmetric if $(a, b) \in R \implies (b, a) \in R$.
Consider $(1, 2)$. Since $1 \leq 2^3 = 8$,$(1, 2) \in R$.
However,for $(2, 1)$,$2 \not\leq 1^3 = 1$. Thus,$(2, 1) \notin R$.
Therefore,$R$ is not symmetric.
$3$. Transitivity: $A$ relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$.
Consider $(3, \frac{3}{2})$ and $(\frac{3}{2}, \frac{6}{5})$.
$3 \leq (\frac{3}{2})^3 = 3.375$ (True) and $\frac{3}{2} \leq (\frac{6}{5})^3 = 1.728$ (True).
However,for $(3, \frac{6}{5})$,$3 \not\leq (\frac{6}{5})^3 = 1.728$.
Therefore,$R$ is not transitive.
Conclusion: The relation $R$ is neither reflexive,nor symmetric,nor transitive.