## NCERT Solutions for Class 11 Maths Chapter 2 – Relations and Functions Ex 2.2

#### Page No 35:

#### Question 1:

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(

*x*,*y*): 3*x*–*y*= 0, where*x*,*y*∈ A}. Write down its domain, codomain and range.#### Answer:

The relation R from A to A is given as

R = {(

*x*,*y*): 3*x*–*y*= 0, where*x*,*y*∈ A}
i.e., R = {(

*x*,*y*): 3*x*=*y*, where*x*,*y*∈ A}
∴R = {(1, 3), (2, 6), (3, 9), (4, 12)}

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

∴Domain of R = {1, 2, 3, 4}

The whole set A is the codomainof the relation R.

∴Codomain of R = A = {1, 2, 3, …, 14}

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

∴Range of R = {3, 6, 9, 12}

#### Page No 36:

#### Question 2:

Define a relation R on the set

**N**of natural numbers by R = {(*x*,*y*):*y*=*x*+ 5,*x*is a natural number less than 4;*x*,*y*∈**N**}. Depict this relationship using roster form. Write down the domain and the range.#### Answer:

R = {(

*x*,*y*):*y*=*x*+ 5,*x*is a natural number less than 4,*x*,*y*∈**N**}
The natural numbers less than 4 are 1, 2, and 3.

∴R = {(1, 6), (2, 7), (3, 8)}

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

∴ Domain of R = {1, 2, 3}

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

∴ Range of R = {6, 7, 8}

#### Question 3:

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(

*x*,*y*): the difference between*x*and*y*is odd;*x*∈ A,*y*∈ B}. Write R in roster form.#### Answer:

A = {1, 2, 3, 5} and B = {4, 6, 9}

R = {(

*x*,*y*): the difference between*x*and*y*is odd;*x*∈ A,*y*∈ B}
∴R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}

#### Question 4:

The given figure 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?

#### Answer:

According to the given figure, P = {5, 6, 7}, Q = {3, 4, 5}

(i) R = {(

*x, y*):*y = x*– 2;*x*∈ P} or R = {(*x, y*):*y = x*– 2 for*x*= 5, 6, 7}
(ii) R = {(5, 3), (6, 4), (7, 5)}

Domain of R = {5, 6, 7}

Range of R = {3, 4, 5}

#### Question 5:

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

{(

*a*,*b*):*a*,*b*∈ A,*b*is exactly divisible by*a*}.
(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

#### Answer:

A = {1, 2, 3, 4, 6}, R = {(

*a*,*b*):*a*,*b*∈ A,*b*is exactly divisible by*a*}
(i) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}

(ii) Domain of R = {1, 2, 3, 4, 6}

(iii) Range of R = {1, 2, 3, 4, 6}

#### Question 6:

Determine the domain and range of the relation R defined by R = {(

*x*,*x*+ 5):*x*∈ {0, 1, 2, 3, 4, 5}}.#### Answer:

R = {(

*x*,*x*+ 5):*x*∈ {0, 1, 2, 3, 4, 5}}
∴ R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}

∴Domain of R = {0, 1, 2, 3, 4, 5}

Range of R = {5, 6, 7, 8, 9, 10}

#### Question 7:

Write the relation R = {(

*x*,*x*^{3}):*x*is a prime number less than 10} in roster form.#### Answer:

R = {(

*x*,*x*^{3}):*x*is a prime number less than 10}
The prime numbers less than 10 are 2, 3, 5, and 7.

∴R = {(2, 8), (3, 27), (5, 125), (7, 343)}

#### Question 8:

Let A = {

*x*,*y*, z} and B = {1, 2}. Find the number of relations from A to B.#### Answer:

It is given that A = {

*x*,*y*, z} and B = {1, 2}.
∴ A × B = {(

*x*, 1), (*x*, 2), (*y*, 1), (*y*, 2), (*z*, 1), (*z*, 2)}
Since

*n*(A × B) = 6, the number of subsets of A × B is 2^{6}.
Therefore, the number of relations from A to B is 2

^{6}.#### Question 9:

Let R be the relation on

**Z**defined by R = {(*a*,*b*):*a*,*b*∈**Z**,*a*–*b*is an integer}. Find the domain and range of R.#### Answer:

R = {(

*a*,*b*):*a*,*b*∈**Z**,*a*–*b*is an integer}
It is known that the difference between any two integers is always an integer.

∴Domain of R =

**Z**
Range of R =

**Z**_{}

^{}