Relations
If A = {a, b, c}, then the number of equivalence relations on A containing (a, b) is
Select the correct option:
Solution
2
- Mandatory Pairs: Any equivalence relation containing (a,b) must contain:
- Diagonal: (a,a),(b,b),(c,c) (Reflexivity)
- Symmetry: (b,a)
- Case Analysis (Partitions of A): Equivalence relations correspond to partitions of the set.
- Partition 1 ({a,b},{c}): R={(a,a),(b,b),(c,c),(a,b),(b,a)}. (Contains (a,b))
- Partition 2 ({a,b,c}): R=A×A. (Contains all 9 pairs). (Contains (a,b))
- Conclusion: There are 2 such relations.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- relations
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
2
- Mandatory Pairs: Any equivalence relation containing (a,b) must contain:
- Diagonal: (a,a),(b,b),(c,c) (Reflexivity)
- Symmetry: (b,a)
- Case Analysis (Partitions of A): Equivalence relations correspond to partitions of the set.
- Partition 1 ({a,b},{c}): R={(a,a),(b,b),(c,c),(a,b),(b,a)}. (Contains (a,b))
- Partition 2 ({a,b,c}): R=A×A. (Contains all 9 pairs). (Contains (a,b))
- Conclusion: There are 2 such relations.
This hard difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of relations. It appeared in the 2025 exam.
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