We have to find equivalence relations on set A and we know that a relation is said to be equivalence relation if the relation is reflexive, symmetric and transitive on set A.

Condition of reflexive relation on set = {1, 2, 3} is ordered pairs (1, 1), (2, 2) and (3, 3)must belongs to the relation.

We have to find such equivalence relation which containing (1, 2)and (2, 1).

Therefore, smallest equivalence relation which containing (1, 2) and (2, 1) on set {1, 2, 3} is R₁ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}.

If we include any one ordered pair outside this relation R₁ like as (1, 3), (2, 3), (3, 1) and (3, 2), then for satisfying equivalence relation conditions gives us the Cartesian product A x A, which is largest relation on set A.

R₂ = R₁ ∪ {(3, 1)} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (3, 1)}

Since, we want to make a equivalence relation.

Then (1, 3) ∈ R₂. (Condition of symmetric relation )

Therefore, R₂ extended to R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (3, 1), (1, 3)}.

Since, (3, 1) ∈ R₂ and (1, 2) ∈ R₂. Then, (3, 2) ∈ R₂, therefore (2, 3) ∈ R₂.Given set A = {1, 2, 3}.

(Conditions of symmetric and transitive relations ).

Therefore, R₂ extended to R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (3, 1), (1, 3), (2, 3), (3, 2)} = A x A which is largest relation defined on A.

Therefore, only possible equivalence relations on set A are R₁ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} and R₂ = A x A.

Hence, total number of equivalence relations on the set {1, 2, 3} containing (1, 2)and (2, 1) are 2.