3. If A={1,2,3}, Find the Elements of the Relation R={(x,y): x=y and x,y∈A} on A – Advance Math Class 10 Exercise 1.3
Complete NCERT/ASSEB Solution with Step-by-Step Explanation, Identity Relation and Roster Form.
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| Identity Relation on Set A |
Introduction
A relation is a set of ordered pairs that satisfy a specified condition. In this question, the relation is defined on the set A={1,2,3} with the condition x=y. Therefore, only those ordered pairs in which both elements are equal will belong to the relation.
Question
Let A={1,2,3} be a set. A relation R on A is defined by
R={(x,y): x=y and x,y∈A}
Find the elements of the relation R.
Given Data
Set: A={1,2,3}
Relation:
R={(x,y): x=y and x,y∈A}
Understanding the Relation
The condition x=y means both elements of every ordered pair must be equal.
- (1,1) ✔ because 1=1
- (2,2) ✔ because 2=2
- (3,3) ✔ because 3=3
- (1,2), (2,3), (3,1), etc. ✘ because x≠y
Hence, only the ordered pairs having equal first and second elements belong to the relation. In the next step, we will determine the complete relation in roster form and present the final answer.
Step-by-Step Solution
The relation is defined by
R={(x,y): x=y and x,y∈A}
where
A={1,2,3}
Step 1: Write the Elements of Set A
The given set contains the following elements:
A={1,2,3}
Since the relation is defined on the set A, both x and y must belong to this set.
Step 2: Apply the Condition x=y
The condition x=y means that the first and second elements of every ordered pair must be equal.
| Ordered Pair | x=y? | Belongs to R? |
|---|---|---|
| (1,1) | ✔ Yes | ✔ Included |
| (1,2) | ✘ No | ✘ Excluded |
| (1,3) | ✘ No | ✘ Excluded |
| (2,1) | ✘ No | ✘ Excluded |
| (2,2) | ✔ Yes | ✔ Included |
| (2,3) | ✘ No | ✘ Excluded |
| (3,1) | ✘ No | ✘ Excluded |
| (3,2) | ✘ No | ✘ Excluded |
| (3,3) | ✔ Yes | ✔ Included |
Step 3: Write the Relation in Roster Form
The ordered pairs satisfying the condition x=y are:
R={(1,1),(2,2),(3,3)}
This relation is known as the Identity Relation on the set A, because every element is related only to itself.
Final Answer
Required Relation
R={(1,1),(2,2),(3,3)}
Hence, the elements of the relation are: (1,1), (2,2), and (3,3).
Important Concepts
1. Relation
A relation is a subset of the Cartesian product of two sets. It consists of ordered pairs that satisfy a specified condition.
2. Ordered Pair
An ordered pair (x,y) belongs to the relation only if it satisfies the given condition.
3. Identity Relation
An identity relation on a set contains only those ordered pairs in which the first and second elements are the same.
I={(a,a):a∈A}
4. Roster Form
Roster form represents a relation by listing all valid ordered pairs inside curly braces.
5. Equality Condition
If the condition is x=y, then only ordered pairs having identical first and second elements are included.
Quick Revision
- ✔ A={1,2,3}
- ✔ Relation is defined by x=y.
- ✔ Only equal ordered pairs satisfy the condition.
- ✔ Required relation: R={(1,1),(2,2),(3,3)}.
- ✔ This is the Identity Relation on the set A.
Exam Tips
- Always identify the condition given in the relation before writing ordered pairs.
- For x=y, include only pairs where both coordinates are equal.
- Exclude every pair for which x≠y.
- Write the final relation in proper roster form using curly braces.
- Remember that the identity relation always relates each element to itself only.
Frequently Asked Questions (FAQs)
1. What is a relation in Mathematics?
A relation is a set of ordered pairs satisfying a specified condition.
2. What is an identity relation?
An identity relation is a relation in which every element is related only to itself.
3. What are the elements of the relation in this question?
The elements are (1,1), (2,2), and (3,3).
4. Why is (1,2) not included?
Because the condition requires x=y, but here 1≠2.
5. What is roster form?
Roster form lists all valid ordered pairs enclosed within curly braces.
6. Is this relation finite?
Yes. Since the set A={1,2,3} contains only three elements, the relation has only three ordered pairs.
7. What is the final answer?
R={(1,1),(2,2),(3,3)}.
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
4. Infinite Relation in Roster Form
5. Relation in Tabular Form, Arrow Diagram and Matrix
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 Textbook – Class 10
- ASSEB Mathematics Textbook
- Relations and Functions – Exercise 1.3
- CBSE Class 10 Mathematics Syllabus
- Standard School Mathematics Reference Books