Editing Disjoint sets (section)

==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>