Check whether the relation R in R defined

$R=\left\{(a, b): a \leq b^{3}\right\}$

It is observed that $\left(\frac{1}{2}, \frac{1}{2}\right) \notin \mathrm{R}$ as $\frac{1}{2}>\left(\frac{1}{2}\right)^{3}=\frac{1}{8}$.

∴ R is not reflexive.

Now,

$(1,2) \in R\left(\right.$ as $\left.1<2^{3}=8\right)$

But,

$(2,1) \notin R\left(\operatorname{as} 2>1^{3}=1\right)$

∴ R is not symmetric.

We have $\left(3, \frac{3}{2}\right),\left(\frac{3}{2}, \frac{6}{5}\right) \in \mathrm{R}$ as $3<\left(\frac{3}{2}\right)^{3}$ and $\frac{3}{2}<\left(\frac{6}{5}\right)^{3}$.

But $\left(3, \frac{6}{5}\right) \notin \mathrm{R}$ as $3>\left(\frac{6}{5}\right)^{3}$

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.