Relations practice · Mixed difficulty

Let $A=\{2,3,4,5\}$ and let $R=\{(2,2),(3,3),(4,4),(5,5),(2,3),(3,2),(3,5),(5,3)\}$ be a relation in $A$. Then $R$ is

A relation $R$ from $\mathbb{C}$ to $\mathbb{R}$ is defined by $x R y$ iff $|x|=y$. Which of the following is correct?

Consider the following statements

I. Let $n(A)=m$ and $n(B)=n$. Then the total number of non-empty relations that can be defined from $A$ to $B$ is $2^{{mn}}-1$

II. If ${A}=\{1,2,3\}, {B}=\{3,8\}$, then $({A} \cup {B}) \times({A} \cap {B})$ is equal to $\{(1,3),(2,3),(3,3),(8,3)\}.$

III. If $\left(\frac{x}{2}-1, \frac{y}{9}+1\right)=(2,1)$, then the values of ${x}$ and ${y}$ respectively are 6 and 0.

Choose the correct option.

Let $R$ be a relation on the set of integers given by $a$ $R$ $b$ $\Rightarrow a=2^k b$ for some integer $k.$ Then, $R$ is

The number of equivalence relations in the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is

Set builder form of the relation $R = \left\{ {\left( { - 2, - 7} \right),\left( { - 1, - 4} \right),\left( {0, - 1} \right),\left( {1,\;2} \right),\left( {2,\;5} \right)} \right\}$ is

Let $A=\{1,2,3\}$. Then find the number of relations containing $(1,2)$ and $(1,3)$, which are reflexive and symmetric but not transitive, is

In the set ${N}$ of natural numbers, define the binary operation $*$ by $m * {n}=\operatorname{GCD}({m}, {n}), {m}, {n} \in {N}$. Then, which of the following is true?

I. $*$ is not a binary operation

II. $*$ is a binary operation

III. Inverse of each element of ${N}$ exist

IV. Inverse of each element of ${N}$ does not exist

The relation ‘is not equal to’ is defined on $R$, is

If $n(X)=5$ and $n(Y)=7$, then the number of relations on ${X} \times {Y}$ is $2^{5 {m}}$. The value of '${m}$' is

If a set $A$ has 10 elements, then the ratio of total no. of reflexive relations to total no. of symmetric relations is

Let $X=\{1,2,3,4,5\}$ and $Y=\{1,3,5,7,9\}$. Which of the following is not a relation from $X$ to $Y$ ?

Let $\mathrm{R}$ be a reflexive relation on a set $\mathrm{A}$ and $\mathrm{I}$ be the identity relation on $\mathrm{A}$. Then

If $A$ is the set of even natural numbers less than 10 and $B$ is the set of prime numbers less than 5 , then the number of relations from $A$ to $B$ is

If a relation $R$ is defined on the set $Z$ of integers as follows:

$(a, b) \in R \Leftrightarrow a^2+b^2=25.$

Then, Domain $(R)=$

In the set $A = \left\{ {1,\;2,\;3,\;4,\;5} \right\},$ a relation $R$ is defined by $R = \left\{ {\left( {x,y} \right):x,y \in A\;\mathrm{and\;}x < y} \right\}.$ Then, $R$ is

If $R$ is a relation from a set $A$ to the $\operatorname{set} B$ and $S$ is a relation from $B$ to $C$, then the relation $S \circ R$

The relation $R = \left\{ {\left( {1,\;1} \right),\;\left( {2,\;2} \right),\;\left( {3,\;3} \right)} \right\}$ on the set $\left\{ {1,\;2,\;3} \right\}$ is

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}.$ Then relation is

Which of the following statements is not correct for the relation R defined by aRb, if and only, if b lives within on kilometre from a?

If a set has 13 elements and $R$ is a reflexive relation on $A$ with $n$ elements, then

Let $A=\{1,2,3\}, B=\{1,3,5\}$. If relation $R$ from $A$ to $B$ is given by $\{(1,3),(2,5),(3,3)\}$, then $R^{-1}$ is

Let $P=\left\{(x, y):\left|x^2+y^2\right|=1, x, y \in R\right\}$. Then $\mathrm{P}$ is

Let a relation $R$ be defined by $R=\{(4,5),(1,4),(4,6),(7,6),(3,7)\}$. The relation $R^{-1} o R$ is given by

If $A = \{ x \in R:0 < x < 1\} \,{\text{and}}\,B = \{ x \in R: - 1 < x < 1\} ,$ then $A \times B$ is the set of all points lying

A survey shows that $73 \%$ of the persons working in an office like coffee, whereas $65 \%$ like tea. If $x$ denotes the percentage of them who like both coffee and tea, then $x$ cannot be

Assume $\mathrm{R}$ and $\mathrm{S}$ are (nonempty) relations in a set $A$. Which of the following relation given below is false

Let $R$ be a relation on the set of integers given by $aRb \Leftrightarrow a = {2^k}.b$ for some integer $k$. Then, $R$ is