en>Qwertyus: link "amount of space" to space complexity

2015-01-12T12:57:09Z

link "amount of space" to space complexity

en>Bgwhite: Do general fixes and cleanup. - using AWB (9863)

2014-01-14T08:55:00Z

Do general fixes and cleanup. - using AWB (9863)

← Older revision		Revision as of 10:55, 14 January 2014
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	~~The writer~~ is ~~recognized~~ by the title of ~~Figures Wunder~~. ~~Playing baseball~~ is the ~~hobby he will never quit doing~~. ~~Managing individuals~~ is ~~his occupation~~. ~~Puerto Rico~~ is ~~exactly where he~~ and ~~his wife live~~.<br><br>~~My web~~-~~site~~ - [http://~~Iiab~~.co/~~diettogoreviews74620 Iiab~~.co]		[[Image:Disjunkte Mengen.svg\|thumb\|Two disjoint sets.]]
			In [[mathematics]], two [[Set (mathematics)\|sets]] are said to be '''disjoint''' if they have no [[element (mathematics)\|element]] in common. Equivalently, 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\|authorlink=Paul Halmos\|publisher=Springer\|year=1960\|isbn=9780387900926\|page=15\|url=http://books.google.com/books?id=x6cZBQ9qtgoC&pg=PA15}}.</ref>
			For example, {1, 2, 3} and {4, 5, 6} are '''disjoint sets.'''

			==Generalizations==
			This definition of disjoint sets can be extended to any [[family of sets]]. A family of sets is '''pairwise disjoint''' or '''mutually disjoint''' if every two sets in the family are disjoint.<ref name="halmos"/>
			For example, the collection of sets {{nowrap\|1={ {1}, {2}, {3}, ... } }} is pairwise disjoint.

			[[Almost disjoint sets]] are sets whose 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=http://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=http://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 their [[intersection (set theory)\|intersections]].

			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=http://books.google.com/books?id=fO3tvd9qjLkC&pg=PA59}}.</ref>

			A family ''F'' of sets is pairwise disjoint if, for every two sets in the family, their intersection is empty.<ref name="halmos"/>
			If the family contains more than one set, this implies that the intersection of the whole family is also empty. However, a family of only one set is pairwise disjoint, regardless of whether that set is empty, and may have a non-empty intersection. Additionally, a family of sets may have an empty intersection without being pairwise disjoint.<ref>{{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=9780495562023\|page=95\|url=http://books.google.com/books?id=jJUs0ZDOOHoC&pg=PA95}}.</ref> For instance, the three sets {{nowrap\|1={ {1, 2}, {2, 3}, {1, 3} } }} have an empty intersection but are not pairwise disjoint. In fact, there are no two disjoint sets in this collection.

			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\|authorlink=Béla Bollobás\|publisher=Cambridge University Press\|year=1986\|isbn=9780521337038\|page=82\|url=http://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. 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}}.</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=http://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=http://books.google.com/books?id=rUudIPZD-B0C&pg=PA21}}.</ref>
			For families of more than 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
			*[[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:Set families]]

en>Dfeuer: /* Relation among other classes */ added fact that P is a proper subclass of EXPTIME to the list

2012-08-22T22:40:18Z

Relation among other classes: added fact that P is a proper subclass of EXPTIME to the list

New page

The writer is recognized by the title of Figures Wunder. Playing baseball is the hobby he will never quit doing. Managing individuals is his occupation. Puerto Rico is exactly where he and his wife live.<br><br>My web-site - [http://Iiab.co/diettogoreviews74620 Iiab.co]

PSPACE - Revision history

en>Qwertyus: link "amount of space" to space complexity

en>Bgwhite: Do general fixes and cleanup. - using AWB (9863)

en>Dfeuer: /* Relation among other classes */ added fact that P is a proper subclass of EXPTIME to the list