How many equivalence relations on a 4-element set have exactly 2 equivalence classes?

Correct answer: 7

The correct answer is D. Every equivalence relation corresponds exactly to a partition of the set into nonempty, unlabeled equivalence classes. We must partition four distinct elements into exactly two classes. There are two possible class-size patterns. For sizes 1 and 3, choose the single element; this gives 4 partitions. For sizes 2 and 2, choose two elements for one class, but choosing the other two describes the same partition, so the count is C(4,2)/2 = 3. Therefore the total is 4 + 3 = 7. Hence option B, not D, is correct. The key warning is that equivalence classes are unlabeled: swapping their names does not create a new relation.