Editing Partition of a set (section)

== Partitions and equivalence relations ==
For any [[equivalence relation]] on a set ''X'', the set of its [[equivalence class]]es is a partition of ''X''. Conversely, from any partition ''P'' of ''X'', we can define an equivalence relation on ''X'' by setting {{nowrap|''x'' ~ ''y''}} precisely when ''x'' and ''y'' are in the same part in ''P''. Thus the notions of equivalence relation and partition are essentially equivalent.{{sfn|Schechter|1997|p=54}}

The [[axiom of choice]] guarantees for any partition of a set ''X'' the existence of a subset of ''X'' containing exactly one element from each part of the partition. This implies that given an equivalence relation on a set one can select a [[representative (mathematics)|canonical representative element]] from every equivalence class.