Editing Disjoint sets

{{Short description|Sets with no element in common}}
{{About|the mathematical concept|the data structure|Disjoint-set data structure}}
[[Image:Disjunkte Mengen.svg|thumb|Two disjoint sets]]

In [[set theory]] in [[mathematics]] and [[Logic#Formal logic|formal logic]], two [[Set (mathematics)|sets]] are said to be '''disjoint sets''' if they have no [[element (mathematics)|element]] in common. Equivalently, two disjoint sets are sets whose [[intersection (set theory)|intersection]] is the [[empty set]].<ref name="halmos">{{citation|title=Naive Set Theory|series=[[Undergraduate Texts in Mathematics]]|first=P. R.|last=Halmos|author-link=Paul Halmos|publisher=Springer|year=1960|isbn=9780387900926|page=15|url=https://books.google.com/books?id=x6cZBQ9qtgoC&pg=PA15}}.</ref> For example, {1, 2, 3} and {4, 5, 6} are ''disjoint sets,'' while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint.

==Generalizations==
[[File:Disjoint sets.svg|thumb|A disjoint collection of sets]]

This definition of disjoint sets can be extended to [[family of sets|families of sets]] and to [[indexed family|indexed families]] of sets. 
By definition, a collection of sets is called a ''[[family of sets]]'' (such as the [[power set]], for example). In some sources this is a set of sets, while other sources allow it to be a [[multiset]] of sets, with some sets repeated.
An {{em|[[indexed family]] of sets}} <math>\left(A_i\right)_{i \in I},</math> is by definition a set-valued [[Function (mathematics)|function]] (that is, it is a function that assigns a set <math>A_i</math> to every element <math>i \in I</math> in its domain) whose domain <math>I</math> is called its {{em|[[index set]]}} (and elements of its domain are called {{em|indices}}).

There are two subtly different definitions for when a family of sets <math>\mathcal{F}</math> is called '''pairwise disjoint'''. According to one such definition, the family is disjoint if each two sets in the family are either identical or disjoint. This definition would allow pairwise disjoint families of sets to have repeated copies of the same set. According to an alternative definition, each two sets in the family must be disjoint; repeated copies are not allowed. The same two definitions can be applied to an indexed family of sets: according to the first definition, every two distinct indices in the family must name sets that are disjoint or identical, while according to the second, every two distinct indices must name disjoint sets.<ref name="douglas">{{citation|title=A Transition to Advanced Mathematics|first1=Douglas|last1=Smith|first2=Maurice|last2=Eggen|first3=Richard|last3=St. Andre|publisher=Cengage Learning|year=2010|isbn=978-0-495-56202-3|page=95|url=https://books.google.com/books?id=jJUs0ZDOOHoC&pg=PA95}}.</ref> For example, the family of sets {{nowrap|1={ {0, 1, 2}, {3, 4, 5}, {6, 7, 8}, ... } }} is disjoint according to both definitions, as is the family {{nowrap|1={ {..., −2, 0, 2, 4, ...}, {..., −3, −1, 1, 3, 5} } }} of the two parity classes of integers. However, the family <math>(\{n + 2k \mid k\in\mathbb{Z}\})_{n \in \{0, 1, \ldots, 9\}}</math> with 10 members has five repetitions each of two disjoint sets, so it is pairwise disjoint under the first definition but not under the second.

Two sets are said to be [[almost disjoint sets]] if their intersection is small in some sense. For instance, two [[infinite set]]s whose intersection is a [[finite set]] may be said to be almost disjoint.<ref>{{citation|title=Combinatorial Set Theory: With a Gentle Introduction to Forcing|series=Springer monographs in mathematics|first=Lorenz J.|last=Halbeisen|publisher=Springer|year=2011|isbn=9781447121732|page=184|url=https://books.google.com/books?id=NZVb54INnywC&pg=PA184}}.</ref>

In [[topology]], there are various notions of [[separated sets]] with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint [[closure (topology)|closure]]s or disjoint [[neighbourhood (mathematics)|neighborhoods]]. Similarly, in a [[metric space]], [[positively separated sets]] are sets separated by a nonzero [[metric space|distance]].<ref>{{citation|title=Metric Spaces|volume=57|series=Cambridge Tracts in Mathematics|first=Edward Thomas|last=Copson|publisher=Cambridge University Press|year=1988|isbn=9780521357326|page=62|url=https://books.google.com/books?id=egc5AAAAIAAJ&pg=PA62}}.</ref>

==Intersections==
Disjointness of two sets, or of a family of sets, may be expressed in terms of [[intersection (set theory)|intersections]] of pairs of them.

Two sets ''A'' and ''B'' are disjoint if and only if their intersection <math>A\cap B</math> is the [[empty set]].<ref name="halmos"/>
It follows from this definition that every set is disjoint from the empty set,
and that the empty set is the only set that is disjoint from itself.<ref>{{citation|title=Bridge to Abstract Mathematics|series=MAA textbooks|publisher=Mathematical Association of America|first1=Ralph W.|last1=Oberste-Vorth|first2=Aristides|last2=Mouzakitis|first3=Bonita A.|last3=Lawrence|year=2012|isbn=9780883857793|page=59|url=https://books.google.com/books?id=fO3tvd9qjLkC&pg=PA59}}.</ref>

If a collection contains at least two sets, the condition that the collection is disjoint implies that the intersection of the whole collection is empty. However, a collection of sets may have an empty intersection without being disjoint. Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty.<ref name=douglas/> For instance, the three sets {{nowrap|1={ {1, 2}, {2, 3}, {1, 3} } }} have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. 

The empty family of sets is pairwise disjoint.<ref>{{Cite web |title=Is the empty family of sets pairwise disjoint? |url=https://math.stackexchange.com/questions/1211584/is-the-empty-family-of-sets-pairwise-disjoint |access-date=2024-10-10 |website=Mathematics Stack Exchange |language=en}}</ref>

A [[Helly family]] is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the [[closed interval]]s of the [[real number]]s form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint.<ref>{{citation|title=Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability|first=Béla|last=Bollobás|author-link=Béla Bollobás|publisher=Cambridge University Press|year=1986|isbn=9780521337038|page=82|url=https://books.google.com/books?id=psqFNlngZDcC&pg=PA82}}.</ref>

==Disjoint unions and partitions==
A [[partition of a set]] ''X'' is any collection of mutually disjoint non-empty sets whose [[union (set theory)|union]] is ''X''.<ref name="h60-28">{{harvtxt|Halmos|1960}}, p.&nbsp;28.</ref> Every partition can equivalently be described by an [[equivalence relation]], a [[binary relation]] that describes whether two elements belong to the same set in the partition.<ref name="h60-28"/>
[[Disjoint-set data structure]]s<ref>{{Citation |first1=Thomas H. |last1=Cormen |author1-link=Thomas H. Cormen |first2=Charles E. |last2=Leiserson |author2-link=Charles E. Leiserson |first3=Ronald L. |last3=Rivest |author3-link=Ronald L. Rivest |first4=Clifford |last4=Stein |author4-link=Clifford Stein |title=[[Introduction to Algorithms]] |edition=Second |publisher=MIT Press |year=2001 |isbn=0-262-03293-7 |chapter=Chapter 21: Data structures for Disjoint Sets |pages=498–524 }}.</ref> and [[partition refinement]]<ref>{{citation
 | last1 = Paige | first1 = Robert
 | last2 = Tarjan | first2 = Robert E.
 | doi = 10.1137/0216062
 | mr = 917035
 | issue = 6
 | journal = SIAM Journal on Computing
 | pages = 973–989
 | title = Three partition refinement algorithms
 | volume = 16
 | year = 1987| s2cid = 33265037
 }}.</ref> are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two.

A [[disjoint union]] may mean one of two things. Most simply, it may mean the union of sets that are disjoint.<ref>{{citation|title=Discrete Mathematics: An Introduction to Proofs and Combinatorics|first=Kevin|last=Ferland|publisher=Cengage Learning|year=2008|isbn=9780618415380|page=45|url=https://books.google.com/books?id=gSeC4_uEPTUC&pg=PA45}}.</ref> But if two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make them disjoint before forming the union of the modified sets.<ref>{{citation|title=A Basis for Theoretical Computer Science|series=The AKM series in Theoretical Computer Science: Texts and monographs in computer science|first1=Michael A.|last1=Arbib|first2=A. J.|last2=Kfoury|first3=Robert N.|last3=Moll|publisher=Springer-Verlag|year=1981|isbn=9783540905738|page=9}}.</ref> For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set.<ref>{{citation|title=Understanding Formal Methods|first1=Jean François|last1=Monin|first2=Michael Gerard|last2=Hinchey|publisher=Springer|year=2003|isbn=9781852332471|page=21|url=https://books.google.com/books?id=rUudIPZD-B0C&pg=PA21}}.</ref>
For families of more than two sets, one may similarly replace each element by an ordered pair of the element and the index of the set that contains it.<ref>{{citation|first=John M.|last=Lee|title=Introduction to Topological Manifolds|volume=202|series=Graduate Texts in Mathematics|publisher=Springer|edition=2nd|year=2010|isbn=9781441979407|page=64}}.</ref>

==See also==
*[[Hyperplane separation theorem]] for disjoint convex sets
*[[Mutually exclusive events]]
*[[Relatively prime]], numbers with disjoint sets of prime divisors
*[[Separoid]]
*[[Set packing]], the problem of finding the largest disjoint subfamily of a family of sets

==References==
{{reflist|30em}}

==External links==
*{{MathWorld|title=Disjoint Sets|urlname=DisjointSets}}

{{DEFAULTSORT:Disjoint Sets}}
[[Category:Basic concepts in set theory]]
[[Category:Families of sets]]