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Question
Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.
Options
1
2
3
4
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Solution
Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is 2.
Explanation:
Given that A = {1, 2, 3}
An equivalence relation is reflexive, symmetric, and transitive.
The shortest relation that includes (1, 2) is
R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
It contains more than just the four elements (2, 3), (3, 2), (3, 3) and (3, 1).
Now, if (2, 3) ∈ R1, then for the symmetric relation, there will also be (3, 2) ∈ R1. Again, the transitive relation (1, 3) and (3, 1) will also be in R1.
Hence, any relation greater than R1 will be the only universal relation.
Hence, the number of equivalence relations covering (1, 2) is only two.
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