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Question

Let A ={1,2,3}.Then, number of equivalence relations containing (1,2) is
(a)1
(b)2
(c)3
(d)4

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Solution

It is given that A ={1,2,3}.
An equivalence relation is reflexive, symmetric and transitive.
The smallest equivalence relation containing (1,2) is given by,
R1={(1,1),(2,2),(3,3),(1,2),(2,1)}
Now, we are left with only four pairs i.e., (2,3),(3,2),(1,3), and (3,1).
If we add any one pair [say(2,3)] to R1, then for symmetry we must add (3,2).
Also, for transitivity we are required to add (1,3)and (3,1).
Hence, the only equivalence relation (bigger than R1) is the universal relation.
This shows that the total number of equivalence relations containing (1,2) is two.
Thus, the correct answer is (b).


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