Question:
Check whether the relation $R$ on $\mathbf{R}$ defined by $R=\left\{(a, b): a \leq b^{3}\right\}$ is reflexive, symmetric or transitive.
Solution:
Reflexivity:
Since $\frac{1}{2}>\left(\frac{1}{2}\right)^{3}$
$\left(\frac{1}{2}, \frac{1}{2}\right) \notin R$
So, $R$ is not reflexive.
Symmetry:
Since $\left(\frac{1}{2}, 2\right) \in R$,
$\frac{1}{2}<2^{3}$
But $2>\left(\frac{1}{2}\right)^{3}$
$\Rightarrow\left(2, \frac{1}{2}\right) \in R$
So, $R$ is not symmetric.
Transitivity:
Since $(7,3) \in R$ and $\left(3,3^{\frac{1}{3}}\right) \in R$,
$7<3^{3}$ and $3=\left(3^{\frac{1}{3}}\right)^{3}$
But $7>\left(3^{\frac{1}{3}}\right)^{3}$
$\Rightarrow\left(7,3^{\frac{1}{3}}\right) \notin R$
So, $R$ is not transitive.