KC Sinha Mathematics Solution Class 11 Chapter 2 Relations Exercise 2.2

 Exercise 2.2

Page no - 2.25

Type 1

Question 1 

(i) Let $A=\{2,4,8\}$. Define $R=\{(2,4),(4,8),(8,16)\}$. Is $R$ a relation on $A$ ?

(ii) Let $A=\{1,2,4\}, B=\{2,4,6\}$. Define $R=\{(2,2),(4,4),(6,6)\}$. Is $R$ a relation from $A$ to $B$ ?

(iii) Let $A=\{1,2,3\}, B=\{2,4\}$. Define $R=\{(1,2),(2,2),(2,4),(3,4)\}$. Is $R$ a relation from $A$ to $B$ ?

(iv) Find a relation $R$ from $A=\{1,2,3,4,5\}$ to $B=\mid 1,2,4\}$ defined by $x R y \Leftrightarrow x<y$

Page no - 2.26

Question 2

Write $R$ as a set of ordered pairs

(i) Let $R=\{(x, y): 2 x+3 y<10, x, y \in N\}$.

(ii) Let $R=\{(x, y):(x, y) \in Z \times Z,(x+y)(y+2004)+1=0\}$.

(iii) Let $R=\left\{(x, y):(x, y) \in Z \times Z .4 x^{2}+9 y^{2}=36\right\}$.

Question 3

Let $R=\{(x, y): x, y \in W, y=2 x-4\}$. If $(a,-2) \in R$ and $\left(4, b^{2}\right) \in R$. find the relation $R_{1}=\{(a, b)\}$.

Question 4

Find a linear relation between the components of the ordered pairs of the relation $R$ given by

Question 5

$R=\{(0,2),(-1,5),(2,-4), \ldots\}$

$A=\{a: a \in N, 2 \leq a<6\}$

$B=\{b: b \in N, 3<b<7\}$

$R=\{(a, b): a \in A, b \in B, a$ and $b$ are coprime?

[Hint : $a$ and $b$ are coprime iff HCF of $a$ and $b=1]$

Type 2

Question 6

Find the domain and range of the following relations :

(i) $\{(1,2),(1,4),(1,6),(1,8)\}$.

(ii) $\left\{\left(x, x^{3}\right): x\right.$ is a prime number less than 10$\}$.

(iii) $\{(x, y): x \in N, x<5, y=3\}$.

(iv) $\{(x, y): x \in N, y \in N$ and $x+y=10\}$.

(v) $|(x, x+5): x \in\{0,1,2,3,4,5\}|$

Question 7

Find the domain and range of the relation :

(i) $\left\{\left(x, \frac{1}{x}\right): 0<x<4\right.$ and $x$ is a natural number $\}$.

(ii) $(x, y): x, y \in N$ and $2 x+y=10\}$.

Question 8

Let $R$ be the relation on the set $N$ of all natural numbers defined by $a+3 b=12$. Find (i) $R$ (ii) Domain of $R$ (iii) Range of $R$.

Type 3

Question 9

Let $A=\{1,2\}$ and $B=(3,4\}$. Find the number of relations from $A$ to $B$.

Question 10

Let A= $\{x, y\}$ List all relations on A. Also find the number of relations on A.

Question 11

(i) Let $A=\{a, b, c\}, B=\{x, y\}$. Find the total number of relations from $A$ to $B$.

(ii) Let $A=\{x, y, z\}$ and $B=(1,2\}$, find the number of relations from $A$ to $B$.

Type 4

Question 12

Let $R=\{(1,-1),(2,0),(3,1),(4,2),(5,3)\}$. Then

(i) Write $R$ in builder form

(ii) Represent $R$ by arrow diagram.

Question 13

Let A = $\{1,2,3,4\}$ B = $\{1,2,3,4,5\}$ Let R be a relation from A to B defined by 

$a R b \Leftrightarrow a$ divides $b$.

(i) Represent $R$ by lattice.

(ii) Represent $R$ in tabular form.

Page no -2.27

Question 14

Let $A=\{1,2,3\}, B=\{4,5\}$ and $R$ be a relation from $A$ into $B$ given by $R=\{(2,4),(2,5),(3,5)\}$.

Represent $R$ :

(i) in tabular form.

(ii) by arrow diagram.

Question 15

Let $A=\{1,2,3,4,5,6\}$. Define a relation $R$ from $A$ to $A$ by

$R=\{(x, y): y=x+1\}$

Depict this relation using an arrow diagram. Write down the domain, codomain and range of $R$.

Question 16

Write the relation $R=\left\{\left(x, x^{3}\right): x\right.$ is a prime number less than 10 in roster form.

Question 17

Define a relation $R$ on the sct $N$ of all natural numbers by $R=\{(x, y): y=x+5, x$ is a natural number less than 4$\}$.

Depict this relation using

(i) roster form.

(ii) an arrow diagram

Write down its domain and range.

Question 18

Given figure shows a relation $R$ from set $A$ to set $B$. Write this relation in

(i) set builder form (ii) roster form.

(Diagram to be added)