Let A = 1, 2, 3, 4 . The number of equivalence relations on A that co…

The correct answer is Option (3) → 5

Each equivalence relation on A splits A into disjoint equivalence classes, and different splits give different relations. A relation contains (1, 2) exactly when 1 and 2 are in the same class.

So the classes are found by keeping 1 and 2 together and placing 3 and 4:

    {1, 2}, {3}, {4}
    {1, 2, 3}, {4}
    {1, 2, 4}, {3}
    {1, 2}, {3, 4}
    {1, 2, 3, 4}

There is no other way: 3 and 4 are either both with {1, 2}, one of them with {1, 2}, or both outside it (together or apart). Hence there are 5 such equivalence relations.