TITLE: Number of equivalence relations on a finite set
QUESTION [9 upvotes]: I need a hint for computing the number of ways in which all the equivalent classes on a set of $n$ elements can be realized. For example, if the set has 2 elements ${a,b}$, then there are 2 possible partitions: either a and b are related or not.

REPLY [2 votes]: Every equivalence relation in a set induce a partition of that set into non-empty blocs and vice versa every set partition defines a equivalence relation on that set. Number of set partitions i.e number of equivalence relations on a n-set is called Bell number $B_n$.