Editing Partition of a set (section)

{{Short description|Mathematical ways to group elements of a set}}
{{about|grouping elements of a set|partitioning an integer|Integer partition|the partition calculus of sets|Infinitary combinatorics|the problem of partitioning a multiset of integers so that each part has the same sum|Partition problem}}
[[File:Indiabundleware.jpg|thumb|A set of stamps partitioned into bundles: No stamp is in two bundles, no bundle is empty, and every stamp is in a bundle.]]{{Use American English|date = March 2019}}
[[File:Set partitions 5; circles.svg|thumb|The [[Bell number|52]] partitions of a set with 5 elements. A colored region indicates a subset of X that forms a member of the enclosing partition. Uncolored dots indicate single-element subsets. The first shown partition contains five single-element subsets; the last partition contains one subset having five elements.]]
[[File:Genji chapter symbols groupings of 5 elements.svg|thumb|The traditional Japanese symbols for the 54 chapters of the ''[[Tale of Genji]]'' are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and the green symbol is added for reaching 54).<ref>{{citation|contribution=Two thousand years of combinatorics|first=Donald E.|last=Knuth|author-link=Donald Knuth|pages=7–37|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}</ref>]]
In [[mathematics]], a '''partition of a set''' is a grouping of its elements into [[Empty set|non-empty]] [[subset]]s, in such a way that every element is included in exactly one subset.

Every [[equivalence relation]] on a [[Set (mathematics)|set]] defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a [[setoid]], typically in [[type theory]] and [[proof theory]].