A partition of a set S is a collection of non-empty disjoint subsets of S whose union is S. Equivalently, every element of S belongs to exactly one of the subsets in the collection. Given a partition of a set S, an equivalence relation ∼ on S can be obtained by defining a ∼ b if a and b belong to the same subset in the partition. Conversely, given any equivalence relation on S, the equivalence classes partition S.