How many equivalence relations are there in a set?
Equal to the number of distinct partitions of the set, also known as the Bell number.
The number of equivalence relations in a set with a finite set of elements is equal to the number of distinct partitions that it contains, which is also equal to what is known as the Bell number. To compute for the Bell number, one uses Dobinski’â„¢s formula: Bn = (summation from k=0 to ? )(kn/ek!).