5. A={2,3,4,5} and B={3,6,7,10} be Two Sets and a Relation R is Defined as R={(x,y): x Completely Divides y}. Write the Relation R in Tabular Form, Arrow Diagram and Matrix Table – Advance Math Class 10 Exercise 1.3
Complete NCERT/ASSEB Solution with Relation, Tabular Representation, Arrow Diagram and Matrix Table.
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| Arrow Diagram of Relation R |
Introduction
Relations are subsets of the Cartesian product between two sets. In this problem, the relation is defined using the divisibility condition. We have to determine all ordered pairs for which the first element exactly divides the second element. After obtaining the relation, we represent it in tabular form, arrow diagram and matrix form.
Question
Let A={2,3,4,5} and B={3,6,7,10} be two sets. A relation R is defined by
R={(x,y): x completely divides y where x∈A and y∈B}
Represent the relation R by
- Writing the relation R.
- Tabular Form.
- Arrow Diagram.
- Matrix Representation.
Given Data
A = {2,3,4,5}
B = {3,6,7,10}
Relation:
R={(x,y): x completely divides y}
Understanding the Relation
The ordered pair (x,y) belongs to the relation only if x exactly divides y. Therefore we check each element of A with every element of B.
2 divides 6 and 10 ✔
3 divides 3 and 6 ✔
4 divides none ✘
5 divides 10 ✔
Using these observations we obtain the required relation. In the next step, we will write the complete solution, represent the relation in tabular form, draw the arrow diagram, construct the matrix table and present the final answer.
Step-by-Step Solution
The relation is defined by
R={(x,y): x completely divides y}
We check each element of set A with every element of set B. Whenever x exactly divides y, the ordered pair (x,y) belongs to the relation.
Step 1: Form the Relation R
| x | Elements of B Divisible by x | Ordered Pairs |
|---|---|---|
| 2 | 6, 10 | (2,6), (2,10) |
| 3 | 3, 6 | (3,3), (3,6) |
| 4 | None | — |
| 5 | 10 | (5,10) |
R={(2,6),(2,10),(3,3),(3,6),(5,10)}
Tabular Form of the Relation
The following table shows whether an ordered pair belongs to the relation. A value of 1 indicates that the ordered pair belongs to the relation, while 0 indicates that it does not.
| x → y | 3 | 6 | 7 | 10 |
|---|---|---|---|---|
| 2 | 0 | 1 | 0 | 1 |
| 3 | 1 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 1 |
Arrow Diagram
The arrow diagram of the relation is shown below.
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|
| Arrow Diagram of Relation R |
Matrix Representation
The matrix representation of the relation is obtained by replacing each valid ordered pair with 1 and the remaining entries with 0.
| x → y | 3 | 6 | 7 | 10 |
|---|---|---|---|---|
| 2 | 0 | 1 | 0 | 1 |
| 3 | 1 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 1 |
Final Answer
Relation
R={(2,6),(2,10),(3,3),(3,6),(5,10)}
Tabular Form
Obtained as shown above.
Arrow Diagram
Represented above.
Matrix Representation
The matrix of the relation is exactly the same as the tabular representation shown above.
Important Concepts
1. Relation
A relation from set A to set B is any subset of the Cartesian product A × B.
2. Ordered Pair
An ordered pair (x,y) belongs to the relation only if it satisfies the given condition.
3. Tabular Representation
The tabular form uses 1 and 0 to indicate whether an ordered pair belongs to the relation.
4. Arrow Diagram
An arrow diagram visually represents the relation by connecting each element of set A to the related element(s) of set B.
5. Matrix Representation
A matrix representation of a relation contains 1 where the relation exists and 0 elsewhere.
Quick Revision
- ✔ A = {2,3,4,5}
- ✔ B = {3,6,7,10}
- ✔ Relation is based on exact divisibility.
- ✔ Relation: R={(2,6),(2,10),(3,3),(3,6),(5,10)}
- ✔ Element 4 is not related to any element of set B.
- ✔ Arrow diagram, table and matrix represent the same relation in different forms.
Exam Tips
- Always write the condition of the relation before finding ordered pairs.
- Check every element of set A with every element of set B.
- Do not include ordered pairs that do not satisfy the given condition.
- Use 1 for related elements and 0 otherwise in the matrix.
- The arrow diagram, table and matrix should represent the same relation.
Frequently Asked Questions (FAQs)
1. What is a relation?
A relation is a subset of the Cartesian product of two sets.
2. How is the relation defined in this question?
The relation consists of all ordered pairs (x,y) where x exactly divides y.
3. What is the relation R?
R={(2,6),(2,10),(3,3),(3,6),(5,10)}.
4. Why is 4 not related to any element?
None of the elements 3, 6, 7 or 10 is exactly divisible by 4.
5. What is the purpose of an arrow diagram?
It provides a visual representation of the relation between the elements of two sets.
6. What is the matrix representation of a relation?
It is a matrix containing 1 for related ordered pairs and 0 for unrelated ordered pairs.
7. Can a relation contain repeated ordered pairs?
No. Every ordered pair appears only once in a relation.
8. Which chapter does this question belong to?
This question belongs to Relations (Exercise 1.3) of Class 10 Mathematics.
Related Questions – Exercise 1.3
1. Identity Relation and Universal Relation
2. Write Down All Relations on A
3. Find the Elements of a Relation
6. Determine the Inverse Relation
7. Relation Defined by Divisibility by 3
8. Choose Which of the Following are Relations
9. Relation on Natural Numbers
References
- NCERT Mathematics – Class 10
- ASSEB Mathematics Textbook
- Relations and Functions – Exercise 1.3
- CBSE Class 10 Mathematics Syllabus
- Standard School Mathematics Reference Books