Editing Partition of a set (section)

== Definition and notation ==
A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets<ref>{{Cite book|last=Halmos|first=Paul|title=Naive Set Theory R.|publisher=Springer|year=1960|isbn=9780387900926|page=28|url=https://books.google.com/books?id=x6cZBQ9qtgoC&pg=PA28}}</ref> (i.e., the subsets are nonempty mutually [[disjoint sets]]).

Equivalently, a [[family of sets]] ''P'' is a partition of ''X'' if and only if all of the following conditions hold:<ref>{{cite book|last=Lucas|first=John F.|title=Introduction to Abstract Mathematics|publisher=Rowman & Littlefield|year=1990|isbn=9780912675732|page=187|url=https://books.google.com/books?id=jklsb5JUgoQC&pg=PA187}}</ref>
*The family ''P'' does not contain the [[empty set]] (that is <math>\emptyset \notin P</math>).
*The [[union (set theory)|union]] of the sets in ''P'' is equal to ''X'' (that is <math>\textstyle\bigcup_{A\in P} A = X</math>). The sets in ''P'' are said to '''exhaust''' or '''cover''' ''X''.  See also [[collectively exhaustive events]] and [[cover (topology)]].
* The [[intersection (set theory)|intersection]] of any two distinct sets in ''P'' is empty (that is <math>(\forall A,B \in P)\; A\neq B \implies A \cap B = \emptyset</math>). The elements of ''P'' are said to be [[pairwise disjoint]] or mutually exclusive. See also [[mutual exclusivity]].

The sets in <math>P</math> are called the ''blocks'', ''parts'', or ''cells'', of the partition.{{sfn|Brualdi|2004|pp=44–45}}  If <math>a\in X</math> then we represent the cell containing <math>a</math> by <math>[a]</math>.  That is to say, <math>[a]</math> is notation for the cell in <math>P</math> which contains <math>a</math>.  

Every partition <math>P</math> may be identified with an equivalence relation on <math>X</math>, namely the relation <math>\sim_{\!P}</math> such that for any <math>a,b\in X</math> we have <math>a\sim_{\!P} b</math> if and only if <math>a\in [b]</math> (equivalently, if and only if <math>b\in [a]</math>).  The notation <math>\sim_{\!P}</math> evokes the idea that the equivalence relation may be constructed from the partition.  Conversely every equivalence relation may be identified with a partition.  This is why it is sometimes said informally that "an equivalence relation is the same as a partition".  If ''P'' is the partition identified with a given equivalence relation <math>\sim</math>, then some authors write <math>P = X/{\sim}</math>.  This notation is suggestive of the idea that the partition is the set ''X'' divided into cells.  The notation also evokes the idea that, from the equivalence relation one may construct the partition.

The '''rank''' of <math>P</math> is <math>|X|-|P|</math>, if <math>X</math> is [[Finite set|finite]].