Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • Check whether the relation R in R defined by R = {(a,b): a less than or equal to b^3} is reflexive, symmetric or transitive.

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

Answers (1)

\\$\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 $\leq \mathrm{a}^{3}$

\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}

\\\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 $b \leq a^{3}$

\\$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.}

\\$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 $a \leq c^{3}$

\\\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.

Posted by

lovekush

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE launches entrance exam 2025, career guide for parents to plan for students' future
anam-khan

CBSE Class 12 Board Economics Paper Analysis: 'Student-friendly', well balanced with moderate difficulty
anam-khan

CBSE Class 12 Hindi exam 2025 not postponed; special exam for those who miss

Latest Question