10. If A={1,2,3,4,6} and R is a relation on A defined as R={(x,y):y is exactly divisible by x where x,yЄA} then - Advance Math Class 10 - Exercise 1.3

If A = {1, 2, 3, 4, 6} and R is a Relation on A Defined by Divisibility | Exercise 1.3 Question 10

Learn how to find the elements of a relation, its inverse, domain, and range using a simple step-by-step method. This question is important for Class 10 Mathematics and competitive examinations.

📖 Reading Time: 4–5 Minutes 🗓 Updated: July 2026 📚 Relations | Exercise 1.3

Key Takeaways

  • Understand relations defined by divisibility.
  • Learn how to write a relation in roster form.
  • Find the inverse of a relation.
  • Determine the domain and range of a relation.
  • Useful for Class 10, HSLC and competitive exams.
10. If A={1,2,3,4,6} and R is a relation on A defined as R={(x,y):y is exactly divisible by x where x,yЄA} then - Advance Math Class 10 - Exercise 1.3
10. If A={1,2,3,4,6} and R is a relation on A defined as R={(x,y):y is exactly divisible by x where x,yЄA} then - Advance Math Class 10 - Exercise 1.3

Question

Let A = {1, 2, 3, 4, 6} and let R be the relation on A defined by R = {(x, y) : y is exactly divisible by x, where x, y ∈ A} Answer the following:

  • Find all the elements of R.
  • Find the inverse relation R−1.
  • Find the domain and range of R and R−1.

Given

Set,

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

Relation,

R = {(x, y) : y is exactly divisible by x}

Part (i): Find the Elements of R

A pair (x, y) belongs to R only when y is divisible by x.

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) }

Therefore, R contains 12 ordered pairs.

Part (ii): Find the Inverse Relation

To obtain R−1, interchange the first and second elements of every ordered pair.

R−1 = { (1,1), (2,1), (3,1), (4,1), (6,1), (2,2), (4,2), (6,2), (3,3), (6,3), (4,4), (6,6) }

Part (iii): Find d(R), r(R), d(R−1) and r(R−1)

The domain is the set of all first elements of the ordered pairs, while the range is the set of all second elements.

Relation Domain Range
R {1, 2, 3, 4, 6} {1, 2, 3, 4, 6}
R−1 {1, 2, 3, 4, 6} {1, 2, 3, 4, 6}

d(R) = {1, 2, 3, 4, 6}

r(R) = {1, 2, 3, 4, 6}

d(R−1) = {1, 2, 3, 4, 6}

r(R−1) = {1, 2, 3, 4, 6}

Final Answer

(i) The relation R contains 12 ordered pairs.

(ii) The inverse relation is obtained by interchanging the coordinates of every ordered pair.

(iii) The domain and range of both R and R−1 are

{1, 2, 3, 4, 6}

Shortcut Method

Instead of checking every possible ordered pair, simply apply the divisibility rule.

  • ✔ 1 divides every element.
  • ✔ 2 divides 2, 4 and 6.
  • ✔ 3 divides 3 and 6.
  • ✔ 4 divides only 4.
  • ✔ 6 divides only 6.

Combine all valid ordered pairs to obtain the relation R.

Exam Tips

  • ✔ Check divisibility carefully before writing an ordered pair.
  • ✔ To find the inverse relation, simply reverse each ordered pair.
  • ✔ Domain consists of first elements.
  • ✔ Range consists of second elements.
  • ✔ Arrange ordered pairs neatly to avoid mistakes.

Practice Question

Let A = {1, 2, 4, 8}. Define the relation R = {(x, y) : y is divisible by x}. Find the relation, its inverse, domain and range.

Frequently Asked Questions (FAQs)

1. What is a relation?

A relation is a set of ordered pairs that connects elements of one set with elements of another set (or the same set) according to a given rule.

2. What is the inverse of a relation?

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

3. What are the domain and range of a relation?

The domain is the set of all first elements, while the range is the set of all second elements of the ordered pairs.

4. How many ordered pairs are there in this relation?

The given relation contains 12 ordered pairs.

Quick Revision

  • ✔ Set A = {1, 2, 3, 4, 6}
  • ✔ Relation is based on divisibility.
  • ✔ Total ordered pairs = 12.
  • ✔ R⁻¹ is obtained by reversing every ordered pair.
  • ✔ d(R) = r(R) = d(R⁻¹) = r(R⁻¹) = {1, 2, 3, 4, 6}

Summary Table

Concept Answer
Set {1, 2, 3, 4, 6}
Relation Rule y is divisible by x
Number of Ordered Pairs 12
Domain of R {1, 2, 3, 4, 6}
Range of R {1, 2, 3, 4, 6}
Domain of R−1 {1, 2, 3, 4, 6}
Range of R−1 {1, 2, 3, 4, 6}

Conclusion

In this exercise, the relation is formed by checking whether one element exactly divides another. After listing all valid ordered pairs, the inverse relation is obtained by reversing each pair. Since every element of the set appears as both a first and second element, the domain and range of both R and R−1 are the same.

Related Exercise 1.3 Questions

Tags:
Previous Post Next Post