2. If A={1,2}, Write Down All the Relations on A – Advance Math Class 10 Exercise 1.3
Complete NCERT/ASSEB Solution with Cartesian Product, Total Number of Relations and All 16 Relations on Set A.
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| Formula for Total Number of Relations |
Introduction
A relation on a set is any subset of its Cartesian product. Therefore, to write all the relations on a set, we first determine the Cartesian product and then find all possible subsets of it. Since every subset represents a relation, the total number of relations is equal to the number of subsets of the Cartesian product.
Question
If A={1,2} then write down all the relations on A.
Given Data
Given set:
A={1,2}
Step 1: Find the Cartesian Product
The Cartesian product of a set with itself is obtained by pairing every element of the set with every element, including itself.
A×A={(1,1),(1,2),(2,1),(2,2)}
Thus, the Cartesian product contains 4 ordered pairs.
Step 2: Formula for Total Number of Relations
The total number of relations from a set A to itself is equal to the number of subsets of A×A.
|P(A×A)| = 2n²
Here,
- Number of elements in A = 2
- Number of ordered pairs in A×A = 2² = 4
- Total number of relations = 2⁴ = 16
In the next step, we will list all 16 relations on the set A in roster form and present the final answer.
Step-by-Step Solution
The given set is
A={1,2}
The Cartesian product is
A×A={(1,1),(1,2),(2,1),(2,2)}
Step 1: Find the Total Number of Relations
A relation on A is any subset of A×A. Since A×A contains 4 ordered pairs, the number of possible relations is
|P(A×A)| = 2⁴ = 16
Step 2: Write All Relations on A
The sixteen relations on A are as follows:
- ∅
- {(1,1)}
- {(1,2)}
- {(2,1)}
- {(2,2)}
- {(1,1),(1,2)}
- {(1,1),(2,1)}
- {(1,1),(2,2)}
- {(1,2),(2,1)}
- {(1,2),(2,2)}
- {(2,1),(2,2)}
- {(1,1),(1,2),(2,1)}
- {(1,1),(1,2),(2,2)}
- {(1,1),(2,1),(2,2)}
- {(1,2),(2,1),(2,2)}
- {(1,1),(1,2),(2,1),(2,2)} (=A×A)
Step 3: Explanation
Every relation on a set is simply a subset of its Cartesian product. Since the Cartesian product A×A contains four ordered pairs, every possible subset of these four ordered pairs represents a relation.
The number of subsets of a set containing four elements is 2⁴=16. Hence, there are exactly 16 relations on the set A.
Final Answer
Cartesian Product
A×A={(1,1),(1,2),(2,1),(2,2)}
Total Number of Relations
2⁴=16
Therefore, the set A={1,2} has 16 different relations, listed above.
Important Concepts
1. Relation
A relation on a set A is any subset of the Cartesian product A×A.
2. Cartesian Product
The Cartesian product of two sets consists of all possible ordered pairs formed by taking the first element from the first set and the second element from the second set.
A×A={(1,1),(1,2),(2,1),(2,2)}
3. Power Set
Every subset of the Cartesian product represents a relation. Therefore, the number of relations is equal to the number of elements in the power set of A×A.
4. Formula for Number of Relations
Number of Relations = 2|A×A| = 2n²
where n is the number of elements in the set.
5. Universal Relation
The relation containing every ordered pair of A×A is called the Universal Relation.
Quick Revision
- ✔ Given set: A={1,2}.
- ✔ Cartesian product: A×A={(1,1),(1,2),(2,1),(2,2)}.
- ✔ Number of ordered pairs = 4.
- ✔ Total number of relations = 2⁴=16.
- ✔ Every subset of A×A represents a relation.
- ✔ The empty relation and the universal relation are also included among the 16 relations.
Exam Tips
- Always find the Cartesian product before writing the relations.
- Remember that every subset of the Cartesian product is a relation.
- Use the formula 2n² to calculate the total number of relations on a set with n elements.
- Include both the empty relation (∅) and the universal relation (A×A) in the list.
- Write ordered pairs carefully using the correct notation.
Frequently Asked Questions (FAQs)
1. What is a relation on a set?
A relation on a set is any subset of its Cartesian product.
2. What is the Cartesian product of A={1,2}?
The Cartesian product is {(1,1),(1,2),(2,1),(2,2)}.
3. How many relations are possible on A={1,2}?
There are 16 relations because 2⁴=16.
4. Why are there 16 relations?
Since the Cartesian product contains four ordered pairs, it has 2⁴=16 subsets, and each subset is a relation.
5. What is the empty relation?
The empty relation is ∅, which contains no ordered pairs.
6. What is the universal relation?
The universal relation is the entire Cartesian product: {(1,1),(1,2),(2,1),(2,2)}.
7. What is the formula for the total number of relations?
If a set has n elements, then the number of relations on the set is 2n².
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
3. Find the Elements of a Relation
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