6. Determine R^-1 of the relation R given in question 5 above. Also find d(R^-1) and r(R^-1) - Advance Math Class 10 - Exercise 1.3

6. Determine R^-1 of the Relation R Given in Question 5. Also Find d(R^-1) and r(R^-1) – Advance Math Class 10 Exercise 1.3

Complete NCERT Solution with Step-by-Step Explanation of Inverse Relation, Domain, Range and Final Answer.

📖 Reading Time: 6 Minutes 📅 Updated: July 2026 📘 Class 10 Mathematics | Relations | Exercise 1.3

$Inverse Relation R-1 Class 10 Exercise 1.3$

Inverse Relation of R

Introduction

The inverse of a relation is obtained by interchanging the first and second elements of every ordered pair. After finding the inverse relation, we can determine its domain and range by identifying the first and second components of the ordered pairs respectively.

In this exercise, we use the relation obtained in Question 5 and determine its inverse relation, domain, and range in a systematic manner.

Question

Determine the inverse relation

R^-1

of the relation R given in Question 5. Also find

Domain of R^-1, i.e. d(R^-1).
Range of R^-1, i.e. r(R^-1).

Given Data

Set A = {2,3,4,5}

Set B = {3,6,7,10}

The relation obtained in Question 5 is:

R = {(2,6), (2,10), (3,3), (3,6), (5,10)}

We have to determine the inverse relation and then find its domain and range.

What is an Inverse Relation?

The inverse relation is formed by reversing every ordered pair of the original relation. If

(x,y) ∈ R

then

(y,x) ∈ R^-1

In the next step, we shall interchange the coordinates of each ordered pair of R, obtain R^-1, and then determine its domain and range.

Step-by-Step Solution

The inverse relation is obtained by interchanging the first and second elements of every ordered pair of the given relation.

R = {(2,6), (2,10), (3,3), (3,6), (5,10)}

Step 1: Find the Inverse Relation R^-1

Interchange the coordinates of every ordered pair as shown below.

Ordered Pair in R	Ordered Pair in R^-1
(2,6)	(6,2)
(2,10)	(10,2)
(3,3)	(3,3)
(3,6)	(6,3)
(5,10)	(10,5)

R^-1 = {(6,2), (10,2), (3,3), (6,3), (10,5)}

Step 2: Find the Domain d(R^-1)

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

d(R^-1) = {3,6,10}

Step 3: Find the Range r(R^-1)

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

r(R^-1) = {2,3,5}

Step 4: Verification

✔ Every ordered pair of R has been reversed correctly.
✔ The domain of R^-1 is the range of R.
✔ The range of R^-1 is the domain of R.

Final Answer

Required	Answer
Inverse Relation	{(6,2), (10,2), (3,3), (6,3), (10,5)}
d(R^-1)	{3,6,10}
r(R^-1)	{2,3,5}

Important Note

The original article contains a mistake. It lists d(R^-1) = {2,3,5} and r(R^-1) = {3,6,10}, which are interchanged.

For the inverse relation R^-1 = {(6,2), (10,2), (3,3), (6,3), (10,5)}, the correct values are:

d(R^-1) = {3,6,10}

r(R^-1) = {2,3,5}

Important Concepts

1. Inverse Relation

The inverse of a relation is obtained by interchanging the first and second elements of every ordered pair. If (x,y) ∈ R, then (y,x) ∈ R^-1.

2. Domain

The domain of a relation is the set of all first elements appearing in its ordered pairs.

3. Range

The range of a relation is the set of all second elements appearing in its ordered pairs.

4. Relationship Between a Relation and Its Inverse

For every relation,

Domain of R = Range of R^-1
Range of R = Domain of R^-1

This property helps verify whether the inverse relation has been obtained correctly.

Quick Revision

✔ Reverse every ordered pair to obtain the inverse relation.
✔ R = {(2,6), (2,10), (3,3), (3,6), (5,10)}
✔ R^-1 = {(6,2), (10,2), (3,3), (6,3), (10,5)}
✔ d(R^-1) = {3,6,10}
✔ r(R^-1) = {2,3,5}
✔ Domain and range interchange after taking the inverse relation.

Exam Tips

Always interchange the coordinates of every ordered pair carefully.
Remember that the domain becomes the range after taking the inverse relation.
Similarly, the range becomes the domain.
Do not confuse the original relation with its inverse.
Verify your answer using the property: d(R^-1) = r(R) and r(R^-1) = d(R).

Frequently Asked Questions (FAQs)

1. What is an inverse relation?

An inverse relation is formed by reversing every ordered pair of the original relation.

2. How do we find R^-1?

Replace every ordered pair (x,y) by (y,x).

3. What is the inverse relation in this question?

R^-1 = {(6,2), (10,2), (3,3), (6,3), (10,5)}.

4. What is d(R^-1)?

The domain of the inverse relation is {3,6,10}.

5. What is r(R^-1)?

The range of the inverse relation is {2,3,5}.

6. Why do domain and range interchange?

Because every ordered pair is reversed, the first elements become second elements and vice versa.

7. How can I check whether my inverse relation is correct?

Verify that d(R^-1) = r(R) and r(R^-1) = d(R).

8. Which chapter does this question belong to?

This question belongs to the chapter Relations in Class 10 Mathematics (Exercise 1.3).

References

NCERT Mathematics – Class 10
ASSEB (SEBA) Mathematics Textbook
Relations and Functions – Exercise 1.3
CBSE Mathematics Curriculum
Standard School Mathematics Reference Books

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