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Question
Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
Options
1
2
3
4
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Solution
Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is 1.
Explanation:
Since the relation R is reflexive.
Again, since the relation R is symmetric.
Hence, (1, 2), (2, 1) ∈ R and (1, 3), (3, 1) ∈ R
But the relation R is not transitive.
Hence, (3, 1), (1, 2) ∈ R but (3, 2) ∈ R
Now if we take any of the elements (3, 2) and (2, 3) in R, then R becomes transitive.
Hence, the number of required relations is one.
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