Relations
The minimum number of ordered pairs that must be added to the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3} to make it an equivalence relation is
Select the correct option:
Solution
7
- Reflexive Closure: Add (1,1),(2,2),(3,3) [3 pairs].
- Symmetric Closure: Add (2,1),(3,2) [2 pairs].
- Transitive Closure:
- Since we have (1,2) and (2,3), we must add (1,3).
- By symmetry, once we add (1,3), we must add (3,1). [2 pairs].
- Verify: The resulting relation is {(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)}, which is A×A (the largest equivalence relation).
- Total Added: 3+2+2=7.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- relations
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
7
- Reflexive Closure: Add (1,1),(2,2),(3,3) [3 pairs].
- Symmetric Closure: Add (2,1),(3,2) [2 pairs].
- Transitive Closure:
- Since we have (1,2) and (2,3), we must add (1,3).
- By symmetry, once we add (1,3), we must add (3,1). [2 pairs].
- Verify: The resulting relation is {(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)}, which is A×A (the largest equivalence relation).
- Total Added: 3+2+2=7.
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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