The number of equivalence relations in the set { 1, 2, 3} containing (1, 2) and (2, 1) is

The smallest equivalence relation R1 containing (1, 2) and (2, 1) is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Now, we are left with only 4 pairs namely (2, 3), (3, 2), (1, 3) and (3, 1). If we add any one, say (2, 3) to RP then for symmetry we must add (3, 2) also and now for transitivity we are forced to add (1, 3) and (3, 1 ). Thus, the only equivalence relation bigger than R1 is the universal relation. This shows that the total number of equivalence relations containing (1, 2) and (2, 1) is two