How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all? Justify your answer.

Let A = {1, 2, 3}

A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

Now, any relation on a set A is subset of A × A.

And any equivalence relation will be reflexive, symmetric and transitive.

Now, Let B be any equivalence relation on A, containing {1, 2, 3}

Now, As B is equivalence relation on A containing (1, 2) and (2, 1), by reflexive property,

(1, 1) ∈ B

(2, 2) ∈ B

(3, 3) ∈ B

Therefore,

An equivalence relation possible is

{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}

Now, if C is another equivalence relation on A containing (1, 2) and (2, 1) then it should contain B and as well as at least one element from (1, 3), (3, 1), (2, 3), (3, 2)

Now, if C contains (1, 3) it should contain by (3, 1) by symmetry and C already contains (2, 1) therefore it should contain (2, 3) by transitive property and hence it should contain all the 4 elements.

And same case is with (3, 1), (2, 3) and (3, 2) and another possible equivalence relation is

{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

Hence, only two equivalence relations are possible.