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Updated 6 August 2026

NCERT Solutions

Relations and Functions

Madhya Pradesh Board · Class 11 · Mathematics

NCERT Solutions for Relations and Functions — Madhya Pradesh Board Class 11 Mathematics.

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EXERCISE 2.1

1If

\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right)

, find the values of

x

and

y

.Show solution

Since two ordered pairs are equal, their corresponding components are equal.

So,

\frac{x}{3}+1=\frac{5}{3}, \qquad y-\frac{2}{3}=\frac{1}{3}

From the first equation,

\frac{x}{3}=\frac{5}{3}-1=\frac{2}{3} \Rightarrow x=2

From the second equation,

y=\frac{1}{3}+\frac{2}{3}=1

Therefore, x = 2 and y = 1.

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2If the set

A

has 3 elements and the set

B = \{3, 4, 5\}

, then find the number of elements in

(A \times B)

.Show solution

If set

A

has 3 elements and set

B

has 3 elements, then

n(A\times B)=n(A)\cdot n(B)=3\cdot 3=9.

So, the number of elements in

A\times B

is 9.

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3If

G = \{7, 8\}

and

H = \{5, 4, 2\}

, find

G \times H

and

H \times G

.Show solution

Using the cartesian product definition, pair each element of

G=\{7,8\}

with each element of

H=\{5,4,2\}

G\times H=\{(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)\}

Similarly, pair each element of

H

with each element of

G

H\times G=\{(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)\}

Since ordered pairs depend on order, these two products are not equal.

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4(i)If

P = \{m, n\}

and

Q = \{n, m\}

, then

P \times Q = \{(m, n), (n, m)\}

.Show solution

the statement is false. Correct statement:

P\times Q=\{(m,n),(m,m),(n,n),(n,m)\}

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4(ii)If

A

and

B

are non-empty sets, then

A \times B

is a non-empty set of ordered pairs

(x, y)

such that

x \in A

and

y \in B

.Show solution

A

and

B

are non-empty sets, then each element of

A

can be paired with each element of

B

. So

A\times B

contains ordered pairs

(x,y)

with

x\in A

and

y\in B

, and it is non-empty. Hence the statement is true.

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4(iii)If

A = \{1, 2\}

B = \{3, 4\}

, then

A \times (B \cap \phi) = \phi

.Show solution

Since

\phi

is the empty set,

B\cap \phi=\phi.

Then

A\times(B\cap\phi)=A\times\phi=\phi.

So the statement is true.

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5If

A = \{-1, 1\}

, find

A \times A \times A

.Show solution

Form all ordered triples from

A=\{-1,1\}

A\times A\times A=\{(a,b,c):a,b,c\in A\}

Thus the 8 triples are

\{(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),(1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)\}.

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6If

A \times B = \{(a, x), (a, y), (b, x), (b, y)\}

. Find

A

and

B

.Show solution

In a cartesian product, the first elements come from the first set and the second elements come from the second set.

Here the first elements are

a

and

b

, so

A=\{a,b\}.

The second elements are

x

and

y

, so

B=\{x,y\}.

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7Let

A = \{1, 2\}

B = \{1, 2, 3, 4\}

C = \{5, 6\}

and

D = \{5, 6, 7, 8\}

. Verify thatShow solution

We have

B\cap C=\varnothing

because

B=\{1,2,3,4\}

and

C=\{5,6\}

have no common element.
So,

A\times(B\cap C)=A\times\varnothing=\varnothing.

Now

A\times B=\{(1,3),(1,4),(2,3),(2,4)\}

and

A\times C=\{(1,5),(1,6),(2,5),(2,6)\}.

Their intersection has no common ordered pair, so

(A\times B)\cap(A\times C)=\varnothing.

Hence,

A\times(B\cap C)=(A\times B)\cap(A\times C).

Also, since every element of

A\times C

has first element in

A\subseteq B

and second element in

C\subseteq D

, we get

A\times C\subseteq B\times D.

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8Let

A = \{1, 2\}

and

B = \{3, 4\}

. Write

A \times B

. How many subsets will

A \times B

have? List them.Show solution

First write the cartesian product:

A\times B=\{(1,3),(1,4),(2,3),(2,4)\}.

It has 4 elements. The number of subsets of a set with 4 elements is

2^4=16.

So,

A\times B

has 16 subsets.

The subsets are:

\varnothing, \{(1,3)\},\{(1,4)\},\{(2,3)\},\{(2,4)\}, \{(1,3),(1,4)\},\{(1,3),(2,3)\},\{(1,3),(2,4)\}, \{(1,4),(2,3)\},\{(1,4),(2,4)\},\{(2,3),(2,4)\},

\{(1,3),(1,4),(2,3)\},\{(1,3),(1,4),(2,4)\},\{(1,3),(2,3),(2,4)\},\{(1,4),(2,3),(2,4)\},

\{(1,3),(1,4),(2,3),(2,4)\}.

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9Let

A

and

B

be two sets such that

n(A) = 3

and

n(B) = 2

. If

(x, 1), (y, 2), (z, 1)

are in

A \times B

, find

A

and

B

, where

x, y

and

z

are distinct elements.Show solution

Since

(x,1)

(y,2)

, and

(z,1)

are in

A\times B

, the first elements belong to

A

and the second elements belong to

B

.

So the distinct first elements are

x,y,z

, hence

A=\{x,y,z\}.

The second elements are

1

and

2

, hence

B=\{1,2\}.

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10The Cartesian product A × A has 9 elements among which are found (−1, 0) and (0,1). Find the set A and the remaining elements of A × A.Show solution

Since

A\times A

has 9 elements,

A

must have 3 elements because

n(A\times A)=n(A)^2=9 \Rightarrow n(A)=3.

The given ordered pairs

( -1,0)

and

(0,1)

show that

-1,0,1

are elements of

A

.
Thus,

A=\{-1,0,1\}.

Now the remaining elements of

A\times A

are all ordered pairs from this set:

A\times A=\{(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)\}.

So the remaining elements are

\{(-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)\}.

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EXERCISE 2.2

1Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where x, y ∈ A}. Write down its domain, codomain and range.Show solution

Given

R=\{(x,y):3x-y=0,\ x,y\in A\}, \quad A=\{1,2,3,\dots,14\}.

y=3x

. For

y

to lie in

A

, we need

3x\le 14

and

x\in A

.
Thus

x=1,2,3,4

give the pairs

(1,3),(2,6),(3,9),(4,12).

So:
- Domain =

\{1,2,3,4\}

- Codomain =

A=\{1,2,3,\dots,14\}

- Range =

\{3,6,9,12\}

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2Define 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.Show solution

Here

y=x+5

and

x

is a natural number less than 4, so

x=1,2,3

.
Therefore the relation in roster form is

R=\{(1,6),(2,7),(3,8)\}.

So,
- Domain =

\{1,2,3\}

- Range =

\{6,7,8\}

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3A = {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.Show solution

The difference between

x

and

y

must be odd, so one of them must be even and the other odd.

From

A=\{1,2,3,5\}

and

B=\{4,6,9\}

:
-

1

with

4,6

gives odd difference; with

9

gives even difference.
-

2

with

9

gives odd difference.
-

3

with

4,6

gives odd difference; with

9

gives even difference.
-

5

with

4,6

gives odd difference; with

9

gives even difference.

Hence

R=\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\}.

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5Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined byShow solution

From the chapter text, this question is Exercise 2.2, Question 5:

Let

A=\{1,2,3,4,6\}

and

R

be the relation on

A

defined by

R=\{(a,b):a,b\in A,\ b \text{ is exactly divisible by } a\}.

Then:

(i) Roster form

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

\{1,2,3,4,6\}

(iii) Range

\{1,2,3,4,6\}

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6Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.Show solution

Given

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

So the ordered pairs are

(0,5),(1,6),(2,7),(3,8),(4,9),(5,10).

Hence:
- Domain =

\{0,1,2,3,4,5\}

- Range =

\{5,6,7,8,9,10\}

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7Write the relation R = {(x, x³) : x is a prime number less than 10} in roster form.Show solution

The prime numbers less than 10 are

2,3,5,7

.
For each

x

, pair it with

x^3

2^3=8,\ 3^3=27,\ 5^3=125,\ 7^3=343.

Therefore,

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

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8Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.Show solution

n(A)=3

and

n(B)=2

, then

n(A\times B)=3\times 2=6.

The number of relations from

A

B

is the number of subsets of

A\times B

2^6=64.

So, the number of relations is 64.

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9Let 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.

EXERCISE 2.3

1Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

2Find the domain and range of the following real functions:

3A function

f

is defined by

f(x) = 2x - 5

. Write down the values of

4The function '

t

' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by

t(C) = \frac{9C}{5} + 32

5Find the range of each of the following functions.

Miscellaneous Exercise on Chapter 2

1The relation

f

is defined by

f(x) = \begin{cases} x^2, & 0 \leq x \leq 3 \\ 3x, & 3 \leq x \leq 10 \end{cases}

The relation

g

is defined by

g(x) = \begin{cases} x^2, & 0 \leq x \leq 2 \\ 3x, & 2 \leq x \leq 10 \end{cases}

Show that

f

is a function and

g

is not a function.

2If

f(x) = x^2

, find

\frac{f(1,1) - f(1)}{(1,1-1)}

3Find the domain of the function

f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}

4Find the domain and the range of the real function

f

defined by

f(x) = \sqrt{(x-1)}

5Find the domain and the range of the real function

f

defined by

f(x) = |x-1|

6Let

f = \left\{ \left( x, \frac{x^2}{1+x^2} \right) : x \in \mathbf{R} \right\}

be a function from

\mathbf{R}

into

\mathbf{R}

. Determine the range of

f

7Let

f, g : \mathbf{R} \to \mathbf{R}

be defined, respectively by

f(x) = x + 1

g(x) = 2x - 3

. Find

f + g, f - g

and

\frac{f}{g}

8Let

f = \{(1,1), (2,3), (0,-1), (-1,-3)\}

be a function from

\mathbf{Z}

\mathbf{Z}

defined by

f(x) = ax + b

, for some integers

a, b

. Determine

a, b

9Let

\mathbf{R}

be a relation from

\mathbf{N}

\mathbf{N}

defined by

\mathbf{R} = \{(a, b) : a, b \in \mathbf{N} \text{ and } a = b^2\}

. Are the following true?

10Let

\mathbf{A} = \{1, 2, 3, 4\}

\mathbf{B} = \{1, 5, 9, 11, 15, 16\}

and

f = \{(1,5), (2,9), (3,1), (4,5), (2,11)\}

. Are the following true?

11Let

f

be the subset of

\mathbf{Z} \times \mathbf{Z}

defined by

f = \{(ab, a + b) : a, b \in \mathbf{Z}\}

. Is

f

a function from

\mathbf{Z}

\mathbf{Z}

? Justify your answer.

12Let

A = \{9, 10, 11, 12, 13\}

and let

f: A \to \mathbf{N}

be defined by

f(n) =

the highest prime factor of

n

. Find the range of

f

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Frequently Asked Questions

What are the important topics in Relations and Functions for Madhya Pradesh Board Class 11 Mathematics?

Key topics in Relations and Functions include Relations and Functions – Complete Concept Map, Relations and Functions - Full Chapter Concept Hierarchy, Relations and Functions — Complete Concept Map. These are the concepts Madhya Pradesh Board Class 11 examiners draw on most — study them first, then practise related questions.

How to score full marks in Relations and Functions — Madhya Pradesh Board Class 11 Mathematics?

Understand the core concepts first, then work through the 136 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

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This page has free step-by-step NCERT Solutions for every exercise question in Relations and Functions (Madhya Pradesh Board Class 11 Mathematics) — written the way examiners award marks: given, formula, working, answer.

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Relations and Functions

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EXERCISE 2.1

EXERCISE 2.2

EXERCISE 2.3

Miscellaneous Exercise on Chapter 2

Frequently Asked Questions

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