7. A = {3,6,8,9} is a Set and aRb iff a − b is Divisible by 3 for a,b ∈ A – Advance Math Class 10 Exercise 1.3
Complete NCERT Solution with Relation Set, Arrow Diagram, Domain, Range, Inverse Relation and Detailed Explanation.
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| Arrow Diagram of the Relation |
Introduction
A relation on a set is a collection of ordered pairs that satisfy a specified mathematical condition. In this question, the relation is defined by the rule that the difference between two elements must be divisible by 3. We identify all ordered pairs satisfying this condition and then determine the domain, range, and inverse relation.
Question
Let
A = {3, 6, 8, 9}
aRb if and only if a − b is divisible by 3, where a,b ∈ A.
Find:
- (i) Write the relation R as a set and draw the arrow diagram.
- (ii) Find the domain d(R) and range r(R).
- (iii) Find the inverse relation R-1.
Given Data
- Set A = {3,6,8,9}
- Relation: aRb ⇔ (a − b) is divisible by 3.
- Only ordered pairs satisfying this condition belong to the relation.
- We have to determine the relation, arrow diagram, domain, range and inverse relation.
Definition of the Relation
Two elements of set A are related whenever the difference between them is exactly divisible by 3. Therefore, every ordered pair (a,b) satisfying
a − b ≡ 0 (mod 3)
is included in the relation R. In the next step, we shall list all such ordered pairs, draw the arrow diagram, determine the domain and range, and finally obtain the inverse relation.
Step-by-Step Solution
The relation is defined by the rule:
aRb ⇔ (a − b) is divisible by 3
Now check every element of A = {3,6,8,9}. Whenever the difference between two elements is divisible by 3, the corresponding ordered pair belongs to the relation.
Step 1: Write the Relation R
The elements satisfying the given condition are:
R = {(3,3), (3,6), (3,9), (6,3), (6,6), (6,9), (9,3), (9,6), (9,9)}
The element 8 does not appear in the relation because the difference between 8 and any other element of the set is not divisible by 3.
Step 2: Arrow Diagram
The corresponding arrow diagram is shown below.
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| Arrow Diagram of Relation R |
Step 3: Find the Domain d(R)
The domain is the set of all first elements appearing in the ordered pairs.
d(R) = {3,6,9}
Step 4: Find the Range r(R)
The range is the set of all second elements appearing in the ordered pairs.
r(R) = {3,6,9}
Step 5: Find the Inverse Relation
The inverse relation is obtained by interchanging the coordinates of every ordered pair.
R-1 = {(3,3), (6,3), (9,3), (3,6), (6,6), (9,6), (3,9), (6,9), (9,9)}
After arranging the ordered pairs, we observe that the inverse contains exactly the same ordered pairs as the original relation.
R-1 = R
Hence, the relation is symmetric.
Final Answer
| Required | Answer |
|---|---|
| Relation R | {(3,3), (3,6), (3,9), (6,3), (6,6), (6,9), (9,3), (9,6), (9,9)} |
| Domain d(R) | {3,6,9} |
| Range r(R) | {3,6,9} |
| Inverse Relation | R-1 = R |
Important Concepts
1. Relation on a Set
A relation on a set A is any subset of the Cartesian product A × A.
2. Domain of a Relation
The domain is the set of all first elements of the ordered pairs in the relation.
3. Range of a Relation
The range is the set of all second elements of the ordered pairs in the relation.
4. Inverse Relation
The inverse relation is obtained by interchanging the first and second elements of every ordered pair in the relation.
5. Symmetric Relation
A relation is called symmetric if (a,b) ∈ R always implies (b,a) ∈ R. In this question, R-1 = R, so the relation is symmetric.
Quick Revision
- ✔ Relation is defined by a − b being divisible by 3.
- ✔ Only the elements 3, 6, and 9 satisfy the condition with each other.
- ✔ Domain = {3,6,9}.
- ✔ Range = {3,6,9}.
- ✔ The inverse relation is the same as the original relation.
- ✔ Therefore, R-1 = R.
Exam Tips
- Always check the given condition for every possible ordered pair.
- Use the divisibility rule carefully while forming the relation.
- The domain consists of the first elements only.
- The range consists of the second elements only.
- To obtain the inverse relation, simply interchange the coordinates of every ordered pair.
- If R = R-1, then the relation is symmetric.
Frequently Asked Questions (FAQs)
1. What is the relation in this question?
The relation is defined by aRb if and only if a − b is divisible by 3.
2. Why is 8 not included in the relation?
The differences involving 8 are not divisible by 3, so it does not form any ordered pair satisfying the given condition.
3. What is the domain of R?
d(R) = {3,6,9}.
4. What is the range of R?
r(R) = {3,6,9}.
5. How is the inverse relation obtained?
The inverse relation is obtained by interchanging the first and second elements of every ordered pair.
6. Why is R-1 equal to R?
Every ordered pair has its reverse pair in the relation. Hence the inverse relation is exactly the same as the original relation.
7. Is this relation symmetric?
Yes. Since R = R-1, the relation is symmetric.
8. Which chapter does this question belong to?
This question belongs to the chapter Relations in Class 10 Mathematics (Exercise 1.3).
Related Questions – Exercise 1.3
1. Identity Relation and Universal Relation on A
2. Write Down All Relations on A
3. Find the Elements of the Relation R
4. Represent the Relation in Roster Form
5. Relation in Tabular Form, Arrow Diagram and Matrix
6. Determine R-1, Domain and Range
8. Choose Which of the Following Are Relations from A to B
9. Relation R Defined on Natural Numbers
10. Relation on A Defined by Divisibility
References
- NCERT Mathematics – Class 10
- ASSEB (SEBA) Mathematics Textbook
- Relations and Functions – Exercise 1.3
- CBSE Mathematics Curriculum
- Standard School Mathematics Reference Books