1.

Let `A = {1, 2, 3}`. Then number of equivalence relations containing (1, 2) is (A) 1 (B) 2 (C) 3 (D) 4

Answer» Here, `A = {1,2,3}`
`:.` Total possible pairs `= {(1,1)(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}`
`:.` Smallest equivalence relation containing `(1,2), (R_1) = {(1,1),(2,2),(3,3),(1,2),(2,1)}`
Now, if we add `(2,3)`, then we have to add `(3,2)` to make it symmetric.
As `(1,2),(1,3)` are there, we have to add `(1,3)` also to make it transitive.
As we are adding `(1,3)`, we need to add `(3,1)` also to make it symmetric.`:. R_2 = {(1,1)(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}`
These are the `two` equivalence relations are possible.
So, `B` is the correct option.


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