Embibe Experts Solutions for Chapter: Set Theory and Relations, Exercise 1: JEE Main - 25 June 2022 Shift 2

The relation $R=\left\{\left(a,b\right):gcd\left(\mathit{a}\mathit{,}\mathit{b}\right)\mathit{=}1, 2a\ne b, a, b\in \mathrm{\mathbb{Z}}\right\}$ is:

The minimum number of elements that must be added to relation $R=\left\{\left(a,b\right),\left(b,c\right),\left(b,d\right)\right\}$ on the set $\left\{a,b,c,d\right\}$, so that it is an equivalence relation is

Let $R$ be a relation defined on $\mathrm{\mathbb{N}}$ as a $R$ b is $2a+3b$ is a multiple of $5,a,b\in \mathrm{\mathbb{N}}$. Then $R$ is

The minimum number of elements that must be added to the relation $R=(a,b),(b,\mathrm{c})$ on the set $\{\mathrm{a},\mathrm{b},\mathrm{c}\}$ so that it becomes symmetric and transitive is:

Let $R$ be a relation on $N\times N$ defined by $\left(a,b\right)R\left(c,d\right)$  if and only if $ad\left(b-c\right)=bc\left(a-d\right)$. Then $R$ is

Among the relations
$S=\left\{\left(a,b\right):a,b\in R-\left\{0\right\},2+\frac{a}{b}>0\right\}$ and $T=\left\{\left(a,b\right):a,b\in R,a^2-b^2\in Z\right\}$,

Let $R$ be a relation on $\mathrm{ℝ}$, given by $R=\{\left(a,b\right):3a-3b+\sqrt{7}$ is an irrational number $\}$. Then $R$ is

Let $P\left(\mathrm{S}\right)$ denote the power set of $S=\left\{1,2,3,\dots ,10\right\}$. Define the relations $R_1$ and $R_2$ on $P\left(S\right)$ as $A{R}_{1}B$ if $\left(A\cap B^c\right)\cup \left(B\cap A^c\right)=\varphi$ and $A{R}_2 B$ if $A\cup B^c=B\cup A^c,\forall A,B\in P\left(S\right)$ . Then :