Since is reflexive, for all , so the classes cover . It remains to show that for all , either or . Suppose ; then , so by symmetry, and , so by transitivity. Now any satisfies , hence , so and ; by symmetry . Therefore , and the equivalence classes form a partition of .

Conversely, given any partition of , there is an equivalence relation whose equivalence classes are precisely the parts of the partition: define when and lie in the same part.