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

Answer

2

Explanation

It is given that `A={1,2,3}.`
An equivalience relation is reflexive, symmetric and transitive.
The smallest equivalence relation containing `(1,2)` is given by,
`R_(1)={(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 `R_(1)`, then for symmetry we must add (3, 2)
Also, for transitivity we are required to add `(1,3) " and " (3,1)`.
Hence, the only euivalence relation (bigger than `R_(1)`) is the universal relation.
This shows that the total number of equivalence relations containing `(1,2)` is two.