1. If A={1,3}, Write the Identity Relation I:A→A and the Universal Relation on A – Advance Mathematics Class 10 Exercise 1.3
Complete NCERT/ASSEB Solution with Identity Relation, Universal Relation, Cartesian Product and Detailed Explanation.
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| Given Set A={1,3} |
Introduction
In mathematics, a relation is a collection of ordered pairs. Two important relations defined on a set are the Identity Relation and the Universal Relation. The identity relation contains only those ordered pairs in which both elements are the same, whereas the universal relation contains every ordered pair in the Cartesian product of the set with itself.
Question
If A={1,3}, write the Identity Relation I:A→A. Also write the Universal Relation on A.
Given Data
The given set is
A={1,3}
Understanding Identity Relation
An Identity Relation relates every element of a set only to itself. Therefore, only ordered pairs of the form (a,a) are included.
I={(a,a):a∈A}
For the set A={1,3}, the identity relation will contain only the pairs in which both coordinates are identical.
Understanding Universal Relation
A Universal Relation contains every possible ordered pair that can be formed from the elements of the set. In other words, it is exactly equal to the Cartesian product A×A.
Universal Relation = A×A
Since every ordered pair is included, the universal relation is the largest possible relation on the set.
Step-by-Step Solution
The given set is
A={1,3}
Step 1: Find the Identity Relation
An Identity Relation contains only those ordered pairs in which the first and second elements are the same.
The elements of the set are:
1, 3
Therefore, the ordered pairs satisfying x=y are:
(1,1), (3,3)
Hence, the identity relation is
I={(1,1),(3,3)}
Step 2: Find the Universal Relation
The Universal Relation is equal to the Cartesian product A×A, which contains every possible ordered pair formed from the elements of the set.
The possible ordered pairs are:
(1,1), (1,3), (3,1), (3,3)
Therefore,
A×A={(1,1),(1,3),(3,1),(3,3)}
Hence, the universal relation is
U={(1,1),(1,3),(3,1),(3,3)}
Explanation
The identity relation connects every element only with itself. Therefore, only the ordered pairs (1,1) and (3,3) belong to the identity relation.
The universal relation contains every possible ordered pair that can be formed from the given set. Hence, it is exactly equal to the Cartesian product A×A.
Final Answer
Identity Relation
I={(1,1),(3,3)}
Universal Relation
U={(1,1),(1,3),(3,1),(3,3)}
Since the universal relation is the Cartesian product of the set with itself,
U=A×A
Important Concepts
1. Relation
A relation is a set of ordered pairs that satisfies a given condition. A relation from a set A to itself is a subset of the Cartesian product A×A.
2. Identity Relation
An identity relation contains only those ordered pairs in which the first and second elements are the same.
I={(a,a):a∈A}
3. Universal Relation
A universal relation contains every ordered pair of the Cartesian product. Therefore, the universal relation is equal to A×A.
4. Cartesian Product
The Cartesian product of two sets is the set of all possible ordered pairs obtained by pairing each element of the first set with every element of the second set.
5. Ordered Pair
An ordered pair consists of two elements written in a specific order as (a,b). Changing the order changes the ordered pair.
Quick Revision
- ✔ Given set: A={1,3}.
- ✔ Identity Relation: I={(1,1),(3,3)}.
- ✔ Cartesian Product: A×A={(1,1),(1,3),(3,1),(3,3)}.
- ✔ Universal Relation is equal to A×A.
- ✔ Every identity relation is a subset of the universal relation.
Exam Tips
- Remember that the identity relation always relates each element only to itself.
- The universal relation always contains every ordered pair of the Cartesian product.
- First write the Cartesian product before finding the universal relation.
- Do not confuse the identity relation with the universal relation.
- Write all ordered pairs using proper mathematical notation.
Frequently Asked Questions (FAQs)
1. What is an identity relation?
An identity relation is a relation in which every element is related only to itself.
2. What is the identity relation on A={1,3}?
The identity relation is {(1,1),(3,3)}.
3. What is a universal relation?
A universal relation contains every ordered pair in the Cartesian product of a set with itself.
4. What is the universal relation on A={1,3}?
The universal relation is {(1,1),(1,3),(3,1),(3,3)}.
5. What is the Cartesian product of A={1,3}?
The Cartesian product is {(1,1),(1,3),(3,1),(3,3)}.
6. Is every identity relation a universal relation?
No. The identity relation is only a subset of the universal relation.
7. Why is (1,3) not included in the identity relation?
Because the identity relation contains only ordered pairs where both elements are equal.
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
2. Write Down All Relations on A
3. Find the Elements of the Relation R={(x,y):x=y}
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
8. Choose the Relations from A to B
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