Which one of following relations on R is an equivalence relation

English

Gujarati

Hindi

1.Relation and Function

medium

Which one of the following relations on $R$ is an equivalence relation

A

$a\,{R_1}\,b \Leftrightarrow |a| = |b|$

B

$a{R_2}b \Leftrightarrow a \ge b$

C

$a{R_3}b \Leftrightarrow a \ {\rm{ divides }}\ b$

D

$a{R_4}b \Leftrightarrow a < b$

Solution

$a R_1 b \Leftrightarrow|a|=|b|$

$a=b$

$R_1$ is a reflexive symmetric and transitive. ( as we know $a=b$ )

$\therefore(a, a)$ is present

$\therefore(a, b)(b, a)$ will also be present.

So it is an equivalence relation .

Standard 12

Mathematics

Share

Similar Questions

Define a relation $R$ on the interval $\left[0, \frac{\pi}{2}\right)$ by $xRy$ if and only if $\sec ^2 x-\tan ^2 y=1$. Then $R$ is :

medium

(JEE MAIN-2025)

Let $A = \{1, 2, 3\}, B = \{1, 3, 5\}$. $A$ relation $R:A \to B$ is defined by $R = \{(1, 3), (1, 5), (2, 1)\}$. Then ${R^{ – 1}}$ is defined by

normal

Let $A=\{1,2,3, \ldots \ldots .100\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $\mathrm{R} \subset \mathrm{R}_1$ and the number of elements in $\mathrm{R}_1$ is $\mathrm{n}$. Then, the minimum value of $n$ is……………………..

easy

(JEE MAIN-2024)

$R$ is a relation over the set of real numbers and it is given by $nm \ge 0$. Then $R$ is

easy

Let L be the set of all lines in a plane and $\mathrm{R}$ be the relation in $\mathrm{L}$ defined as $\mathrm{R}=\left\{\left(\mathrm{L}_{1}, \mathrm{L}_{2}\right): \mathrm{L}_{1}\right.$ is perpendicular to $\left. \mathrm{L} _{2}\right\}$. Show that $\mathrm{R}$ is symmetric but neither reflexive nor transitive.

medium