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Check whether the relation $R$ in $R$ defined by $S =\left\{(a, b): a \leq b^{3}\right\}$ is reflexive, symmetric or transitive.
Solution
$R =\left\{( a , b ): a \leq b ^{3}\right\}$
It is observed that $\left(\frac{1}{2}, \frac{1}{2}\right) \notin R ,$ since, $\frac{1}{2}>\left(\frac{1}{2}\right)^{3}$
$\therefore R$ is not reflexive.
Now, $(1,2)\in R($ as $1<2^{3}=8)$
But, $(2,1)\notin R$ $($ as $2^{3}>1$ $)$
$\therefore R$ is not symmetric.
We have $\left(3, \frac{3}{2}\right),\left(\frac{3}{2}, \frac{6}{5}\right) \in R,$
since $3<\left(\frac{3}{2}\right)^{2}$ and $\frac{3}{2}<\left(\frac{6}{5}\right)^{3}$
But $\left(3, \frac{6}{5}\right) \notin R$ as $3>\left(\frac{6}{5}\right)^{3}$
$\therefore R$ is not transitive.
Hence, $R$ is neither reflexive, nor symmetric, nor transitive.
