9. A Relation R is Defined on the Set of Natural Numbers N as aRb Where a = b² for a, b ∈ N. Write the Relation R and Also Write R-1 in Set Builder Method – Advance Math Class 10 Exercise 1.3
Step-by-Step NCERT Solution with Roster Form, Inverse Relation, Mathematical Explanation, and Exam-Oriented Notes.
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| Relation R Defined on the Set of Natural Numbers |
Introduction
A relation is a collection of ordered pairs that connects elements of one set with elements of the same or another set according to a specified rule. In this question, the relation is defined on the set of natural numbers N by the condition a = b2. Therefore, every ordered pair (a, b) belongs to the relation only when the first element is the square of the second element.
Our objective is to write the relation R in roster form and then determine its inverse relation R-1 in set-builder form. The inverse relation is obtained by interchanging the first and second elements of every ordered pair in the relation.
Question
A relation R is defined on the set of natural numbers N as
aRb ⇔ a = b2, where a, b ∈ N
Write the relation R. Also write R-1 in set-builder method.
Given Data
- Set: N (Natural Numbers)
- Relation: aRb ⇔ a = b2
- Condition: a and b belong to the set of natural numbers.
- Required:
- Write the relation R in roster form.
- Write the inverse relation R-1 in set-builder form.
Step 1: Write the Relation R
The relation is defined by the condition a = b2, where both a and b are natural numbers. Therefore, every ordered pair (a,b) satisfying this condition belongs to the relation.
Set Builder Form
R = {(a, b) : a = b2, a, b ∈ N}
Now substitute the natural numbers one by one.
| b | a = b2 | Ordered Pair |
|---|---|---|
| 1 | 1 | (1,1) |
| 2 | 4 | (4,2) |
| 3 | 9 | (9,3) |
| 4 | 16 | (16,4) |
| 5 | 25 | (25,5) |
| ⋮ | ⋮ | ⋮ |
Relation in Roster Form
R = {(1,1), (4,2), (9,3), (16,4), (25,5), …}
Step 2: Find the Inverse Relation R-1
The inverse of a relation is obtained by interchanging the first and second elements of every ordered pair of the relation.
Rule for Inverse Relation
If (a,b) ∈ R, then (b,a) ∈ R-1
Interchanging every ordered pair gives
R-1 = {(1,1), (2,4), (3,9), (4,16), (5,25), …}
Hence, the inverse relation in set-builder form is
R-1 = {(a,b) : b = a2, a,b ∈ N}
Final Answer
Relation R:
R = {(1,1), (4,2), (9,3), (16,4), (25,5), …}
Inverse Relation R-1 (Set Builder Form):
R-1 = {(a,b) : b = a2, a,b ∈ N}
Important Concepts
Relation
A relation is a set of ordered pairs that satisfies a given mathematical condition between two elements.
Roster Form
In roster form, all ordered pairs satisfying the given condition are written explicitly inside curly braces.
Inverse Relation
The inverse relation is obtained by interchanging the first and second elements of every ordered pair.
Set Builder Form
Set builder form represents a relation by specifying the rule that every ordered pair satisfies.
Quick Revision
- ✔ Relation is defined by a = b2.
- ✔ Every ordered pair satisfies the given condition.
- ✔ Roster form lists all ordered pairs.
- ✔ Inverse relation is obtained by swapping the coordinates.
- ✔ Inverse relation is written as R-1 = {(a,b) : b = a2, a,b ∈ N}.
Exam Tips
- Write the given relation in set-builder form before converting it into roster form.
- Generate ordered pairs carefully by substituting natural numbers.
- While finding the inverse relation, simply interchange the coordinates of every ordered pair.
- Remember that (a,b) ∈ R ⇒ (b,a) ∈ R-1.
- Always write the final answer in the form asked in the question.
Frequently Asked Questions (FAQs)
1. What is a relation?
A relation is a collection of ordered pairs connecting elements according to a specified rule.
2. What is the given relation?
The relation is defined by a = b2, where a and b are natural numbers.
3. What is the roster form of R?
R = {(1,1), (4,2), (9,3), (16,4), (25,5), …}
4. What is an inverse relation?
An inverse relation is obtained by reversing every ordered pair of the original relation.
5. What is the inverse relation in set-builder form?
R-1 = {(a,b) : b = a2, a,b ∈ N}
6. How do you obtain R-1?
Replace every ordered pair (a,b) in R with (b,a).
7. Is every relation invertible?
Yes. Every relation has an inverse relation obtained by interchanging the coordinates of its ordered pairs.
8. Which chapter does this question belong to?
This question belongs to the chapter Relations in Class 10 Mathematics (Exercise 1.3).
Summary Table
| Particular | Result |
|---|---|
| Relation Definition | a = b2 |
| Set | Natural Numbers (N) |
| Relation R | {(1,1), (4,2), (9,3), (16,4), (25,5), …} |
| Inverse Relation R-1 | {(1,1), (2,4), (3,9), (4,16), (5,25), …} |
| Set Builder Form of R-1 | {(a,b) : b = a2, a,b ∈ N} |
Conclusion
The given relation is defined on the set of natural numbers by the condition a = b2. By substituting natural numbers one by one, we obtain the ordered pairs of the relation in roster form. The inverse relation is then found by interchanging the first and second elements of every ordered pair.
Hence, the required relation and its inverse are:
Final Answer
R = {(1,1), (4,2), (9,3), (16,4), (25,5), …}
and
R-1 = {(a,b) : b = a2, a,b ∈ N}
Related Questions – Exercise 1.3
1. If A = {1, 3}, then write the identity relation and the universal relation on A.
2. If A = {1,2}, then write down all the relations on A.
3. Find the elements of the relation R={(x,y):x=y and x,y∈A}.
4. Represent the relation R={(a,b):a−b>2} in roster form.
5. Write the relation in tabular form, arrow diagram and matrix.
6. Determine R-1, domain and range.
7. Write R, draw the arrow diagram and find domain and range.
8. Choose which of the following are relations from A to B.
10. If A={1,2,3,4,6} and R={(x,y):y is exactly divisible by x}, then…
References
- NCERT Mathematics – Class 10
- SEBA/ASSEB Mathematics Textbook
- Relations and Functions – Exercise 1.3
- CBSE Mathematics Curriculum
- Standard School Mathematics Textbooks