Question

Check whether the relative $R$ in $\text{R}$ defined by $R=\left\{\right(a,b):a\le b^3\}$ is reflexive, symmetric or transitive.

Answer 1

$R=\left\{\right(a,b):a\le b^3\}$
It is observed that $\left(\frac{1}{2},\frac{1}{2}\right)\notin R$ as $\frac{1}{2}>{\left(\frac{1}{2}\right)}^3=\frac{1}{8}$.
$\therefore R$ is not reflexive.
Now, $(1,2)\in R(\text{as}1<2^3=8)$
But, $(2,1)\notin R(\text{as}2>1^3)$
$\therefore R$ is not symmetric.
We have $\left(3,\frac{3}{2}\right),\left(\frac{3}{2},\frac{6}{5}\right)\in R$ as $3<{\left(\frac{3}{2}\right)}^3$ and $\frac{3}{2}<{\left(\frac{6}{5}\right)}^3$.
But $(3,\frac{6}{5})\notin R$ as $3>{\left(\frac{6}{5}\right)}^3$.
$\therefore R$ is not transitive.
Hence, $R$ is neither reflexive, nor symmetric, nor transitive.