# NCERT Solutions For Class 11 Maths Chapter 2 Exercise 2.2

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## Chapter 2 Ex.2.2 Question 1

Let $$A = \left\{ {1,2,3, \ldots ,14} \right\}$$. Define a relation $$R$$ from $$A$$ to $$A$$ by $$R = \left\{ {\left( {x,y} \right),3x - y = 0;x,y \in A} \right\}.$$ Write down its domain, co-domain and range.

### Solution

The relation $$R$$ from $$A$$ to $$A$$ is given as $$R = \left\{ {\left( {x,y} \right),3x - y = 0;x,y \in A} \right\}.$$

Thus, $$R = \left\{ {\left( {x,y} \right),3x = y;x,y \in A} \right\}.$$

Therefore, $$R = \left\{ {\left( {1,3} \right),\left( {2,6} \right),\left( {3,9} \right),\left( {4,12} \right)} \right\}$$

The domain of $$R$$ is the set of all first elements of the ordered pairs in the relation.

Hence, Domain of $$R = \left\{ {1,2,3,4} \right\}$$

The whole set $$A$$ is the co-domain of the relation $$R.$$

Therefore, Co-domain of $$R = A = \left\{ {1,2,3, \ldots ,14} \right\}$$

The range of $$R$$ is the set of all second elements of the ordered pairs in the relation.

Therefore, Range of $$R = \left\{ {3,6,9,12} \right\}$$

## Chapter 2 Ex.2.2 Question 2

Define a relation $$R$$ on the set $$N$$ of natural numbers by $$\rm{R} = \{ \left( {x,{\rm{ }}y} \right):{\rm{ }}y = x{\rm{ }} + {\rm{ 5}},$$  $$x$$ is a natural number less than $$4;\;x,\,y \in {\rm{N}}\} .$$

Depict this relationship using roster form. Write down the domain and the range.

### Solution

$$\rm{R} = \{ \left( {x,{\rm{ }}y} \right):{\rm{ }}y = x{\rm{ }} + {\rm{ 5}},$$  $$x$$ is a natural number less than $$4;\;x,\,y \in {\rm{N}}\} .$$

The natural numbers less than $$4$$ are $$1,\; 2,$$ and $$3.$$

Therefore, $${\rm{R = }}\left\{ {\left( {{\rm{1}},{\rm{6}}} \right),\left( {{\rm{2}},{\rm{7}}} \right),\left( {{\rm{3}},{\rm{8}}} \right)} \right\}$$

The domain of $$R$$ is the set of all first elements of the ordered pairs in the relation.

Hence, Domain of $${\rm{R}} = \left\{ {{\rm{1}},{\rm{2}},{\rm{3}}} \right\}$$.

The range of $$R$$ is the set of all second elements of the ordered pairs in the relation.

Therefore, Range of $$R = \left\{ {6,7,8} \right\}$$.

## Chapter 2 Ex.2.2 Question 3

$$A = \left\{ {1,2,3,5} \right\}$$ and $$B = \left\{ {4,6,9} \right\}$$. Define a relation $$R$$ from $$A$$ to $$B$$ by $$R = \left\{ {\left( {x,y} \right):{\text{the difference between }}x{\text{ and }}y{\text{ is odd}};x \in A,y \in B} \right\}.$$ Write $$R$$ in roster form.

### Solution

$$A = \left\{ {1,2,3,5} \right\}$$, $$B = \left\{ {4,6,9} \right\}$$ and

$$R = \left\{ {\left( {x,y} \right):{\text{the difference between }}x{\text{ and }}y{\text{ is odd}};x \in A,y \in B} \right\}$$

Therefore, $$R = \left\{ {\left( {1,4} \right),\left( {1,6} \right),\left( {2,9} \right),\left( {3,4} \right),\left( {3,6} \right),\left( {5,4} \right),\left( {5,6} \right)} \right\}$$

## Chapter 2 Ex.2.2 Question 4

The Fig2.7 shows a relationship between the sets $$P$$ and $$Q.$$ Write this relation

(i) in set-builder form

(ii) in roster form.

What is its domain and range?

### Solution

As per the Fig2.7, $$P = \left\{ {5,6,7} \right\}$$ and $$Q = \left\{ {3,4,5} \right\}$$

(i) $$R = \left\{ {\left( {x,y} \right):y = x - 2;x \in P} \right\}$$ or $$R = \left\{ {\left( {x,y} \right):y = x - 2,for{\rm{ }}\;x = 5,6,7} \right\}$$

(ii) $$R = \left\{ {\left( {5,3} \right),\left( {6,4} \right),\left( {7,5} \right)} \right\}$$

Domain of $$R = \left\{ {5,6,7} \right\}$$

Range of $$R = \left\{ {3,4,5} \right\}$$

## Chapter 2 Ex.2.2 Question 5

Let $$A = \left\{ {1,2,3,4,6} \right\}$$. Let $$R$$ be the relation on $$A$$ defined by

$$\left\{ {\left( {a,b} \right):a,b \in {\rm{A}},b{\text{ is exactly divisible by }}a} \right\}$$

(i) Write $$R$$ in roster form

(ii) Find the domain of $$R$$

(iii) Find the range of $$R.$$

### Solution

It is given that $$A = \left\{ {1,2,3,4,6} \right\}$$ and $$R = \left\{ {\left( {a,b} \right):a,b \in {\rm{A}},b{\text{ is exactly divisible by }}a} \right\}$$

(i) $$R = \left\{ {\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right),\left( {1,4} \right),\left( {1,6} \right),\left( {2,2} \right),\left( {2,4} \right),\left( {2,6} \right),\left( {3,3} \right),\left( {3,6} \right),\left( {4,4} \right),\left( {6,6} \right)} \right\}$$

(ii) Domain of $$R = \left\{ {1,2,3,4,6} \right\}$$

(iii) Range of $$R = \left\{ {1,2,3,4,6} \right\}$$

## Chapter 2 Ex.2.2 Question 6

Determine the domain and range of the relation $$R$$ defined by

$$R = \left\{ {\left( {x,x + 5} \right):x \in \left\{ {0,1,2,3,4,5} \right\}} \right\}$$.

### Solution

It is given that $$R = \left\{ {\left( {x,x + 5} \right):x \in \left\{ {0,1,2,3,4,5} \right\}} \right\}$$

Therefore, $$R = \left\{ {\left( {0,5} \right),\left( {1,6} \right),\left( {2,7} \right),\left( {3,8} \right),\left( {4,9} \right),\left( {5,10} \right)} \right\}$$

Hence,

Domain of $$R = \left\{ {0,1,2,3,4,5} \right\}$$

Range of $$R = \left\{ {5,6,7,8,9,10} \right\}$$

## Chapter 2 Ex.2.2 Question 7

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

### Solution

It is given that $$R = \left\{ {\left( {x,{x^3}} \right):x{\text{ is a prime number less than }}10} \right\}$$

The prime numbers less than $$10$$ are $$2,\; 3,\; 5,$$ and $$7.$$

Therefore, $$R = \left\{ {\left( {2,8} \right),\left( {3,27} \right),\left( {5,125} \right),\left( {7,343} \right)} \right\}$$

## Chapter 2 Ex.2.2 Question 8

Let $$A = \left\{ {x,y,z} \right\}$$ and $$B = \left\{ {1,2} \right\}$$. Find the number of relations from $$A$$ to $$B.$$

### Solution

It is given that $$A = \left\{ {x,y,z} \right\}$$ and $$B = \left\{ {1,2} \right\}$$.

Therefore, $$A \times B = \left\{ {\left( {x,1} \right),\left( {x,2} \right),\left( {y,1} \right),\left( {y,2} \right),\left( {z,1} \right),\left( {z,2} \right)} \right\}$$

Since $$n\left( {A \times B} \right) = 6$$, the number of subsets of $$\left( {A \times B} \right) = {2^6}$$.

Hence, the number of relations from $$A$$ to $$B$$ is $${2^6}$$.

## Chapter 2 Ex.2.2 Question 9

Let $$R$$ be the relation on $$\bf{Z}$$ defined by $$R = \left\{ {\left( {a,b} \right):a,b \in {\bf{Z}},a - b {\text{ is an integer}}} \right\}$$. Find the domain and range of $$R.$$

### Solution

It is given that $$R = \left\{ {\left( {a,b} \right):a,b \in {\bf{Z}},a - b {\text{ is an integer}}} \right\}$$

It is known that the difference between any two integers is always an integer.

Therefore,

Domain of $$R = {\bf{Z}}$$

Range of $$R = {\bf{Z}}$$

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