Editing Equivalence relation (section)

== Fundamental theorem of equivalence relations ==

A key result links equivalence relations and partitions:<ref>Wallace, D. A. R., 1998. ''Groups, Rings and Fields''. p.&nbsp;31, Th. 8. Springer-Verlag.</ref><ref>Dummit, D. S., and Foote, R. M., 2004. ''Abstract Algebra'', 3rd ed. p.&nbsp;3, Prop. 2. John Wiley & Sons.</ref><ref>[[Karel Hrbacek]] & [[Thomas Jech]] (1999) ''Introduction to Set Theory'', 3rd edition, pages 29–32, [[Marcel Dekker]]</ref>
* An equivalence relation ~ on a set ''X'' partitions ''X''.
* Conversely, corresponding to any partition of ''X'', there exists an equivalence relation ~ on ''X''.
In both cases, the cells of the partition of ''X'' are the equivalence classes of ''X'' by ~. Since each element of ''X'' belongs to a unique cell of any partition of ''X'', and since each cell of the partition is identical to an equivalence class of ''X'' by ~, each element of ''X'' belongs to a unique equivalence class of ''X'' by ~. Thus there is a natural [[bijection]] between the set of all equivalence relations on ''X'' and the set of all partitions of ''X''.