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

$\bg_white \\\mathrm{R}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a} \leq \mathrm{b}^{3}\right\} \\ \text{Here R is set of real numbers}\\ \text{Hence both a and b are real numbers.} \\\\ 1. \text{If the relation is reflexive, then }(a, a) \in R \ \ i.e., \ a \leq \mathrm{a}^{3}$

\bg_white \begin{aligned} &\text { For } \mathrm{a}=1, \mathrm{a}^{3} \leq 1 \Rightarrow 1 \leq 1\\ &\text { For } a=2, a^{3} \leq 8 \Rightarrow 2 \leq 8\\ &\text { For } a=\frac{1}{2}, a^{3} \geq \frac{1}{8}\\ &\text { Hence a } \leq \mathrm{a}^{3} \text { is not true for all values of a } \end{aligned}

$\bg_white \\\text{So, the given relation is not reflexive.}\\ \text{To check whether symmetric or not: } \\ \text{If } (\mathrm{a}, \mathrm{b}) \in \mathrm{R} \text{ then }(\mathrm{b}, \mathrm{a}) \in \mathrm{R} \\ If a \leq b^{3} then b \leq a^{3}$

$\bg_white \\For a=2, b=3,2<2^{3}, 3<2^{3} \\ For \mathrm{a}=2, \mathrm{b}=9,2<9^{3}, 9>2^{3} \\ since \mathrm{b} \leq \mathrm{a}^{3} \ is not true for all values of a and \mathrm{b} \\ \text{Hence the given relation is not symmetric.}$

$\bg_white \\To check whether transitive or not: \\ If (\mathrm{a}, \mathrm{b}) \in \mathrm{R} \text{ and } (\mathrm{b}, \mathrm{c}) \in \mathrm{R} then (\mathrm{a}, \mathrm{c}) \in \mathrm{R} \\ If a \leq b^{3} and b \leq c^{3} then a \leq c^{3} \\ For a=1, b=2, c=3, b^{3}=8, c^{3}=27 \Rightarrow a \leq b^{3}, b \leq c^{3} and a \leq c^{3}$

$\bg_white \\\text{For }a=3, b=\frac{3}{2}, c=\frac{4}{3}, b^{3}=\left(\frac{3}{2}\right)^{3}=3.375, c^{3}=\left(\frac{4}{3}\right)^{3}=2.37 \Rightarrow a \leq b^{3}, b \leq c^{3} and a \geq c^{3} \\ since if a \leq \mathrm{b}^{3}, \mathrm{b} \leq \mathrm{c}^{3} and a \leq \mathrm{c}^{3} is not true for all values of a, b, c. \\ Hence, the given relation is not transitive. \\ \therefore the given relation is neither reflexive, symmetric or transitive.$

## Related Chapters

### Preparation Products

##### JEE Main Rank Booster 2021

This course will help student to be better prepared and study in the right direction for JEE Main..

₹ 13999/- ₹ 9999/-
##### Rank Booster NEET 2021

This course will help student to be better prepared and study in the right direction for NEET..

₹ 13999/- ₹ 9999/-
##### Knockout JEE Main April 2021 (Easy Installments)

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 4999/-
##### Knockout NEET May 2021

An exhaustive E-learning program for the complete preparation of NEET..

₹ 22999/- ₹ 14999/-