View source for Set (mathematics)

{{Short description|Collection of mathematical objects}}
[[File:Example of a set.svg|thumb|A set of polygons in an [[Euler diagram|Euler diagram]]]]
[[File:Example of a set rearranged.svg|thumb|This set equals the one depicted above since both have the very same elements.]]

A '''set''' is the [[Mathematical model|mathematical model]] for a collection of different<ref name="Cantor"/> things;<ref name="JainAhmad1995">{{cite book|author1=P. K. Jain|author2=Khalil Ahmad|author3=Om P. Ahuja|title=Functional Analysis|url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA1|year=1995|publisher=New Age International|isbn=978-81-224-0801-0|page=1}}</ref><ref name="Goldberg1986">{{cite book|author=Samuel Goldberg|title=Probability: An Introduction|url=https://books.google.com/books?id=CmzFx9rB_FcC&pg=PA2|date=1 January 1986|publisher=Courier Corporation|isbn=978-0-486-65252-8|page=2}}</ref><ref name="CormenCormen2001">{{cite book|author1=Thomas H. Cormen|author2=Charles E Leiserson|author3=Ronald L Rivest|author4=Clifford Stein|title=Introduction To Algorithms|url=https://books.google.com/books?id=NLngYyWFl_YC&pg=PA1070|year=2001|publisher=MIT Press|isbn=978-0-262-03293-3|page=1070}}</ref> a set contains ''[[Element (mathematics)|elements]]'' or ''members'', which can be [[Philosophy:Mathematical object|mathematical object]]s of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/n11/mode/2up 1]}}  The set with no elements is the [[Empty set|empty set]]; a set with a single element is a [[Singleton (mathematics)|singleton]].  A set may have a finite number of elements or be an [[Infinite set|infinite set]]. 

Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are [[Equality (mathematics)|equal]] (they are the same set).<ref name="Stoll">{{Cite book |last=Stoll |first=Robert |title=Sets, Logic and Axiomatic Theories |year=1974 |publisher=W. H. Freeman and Company |pages= [https://archive.org/details/setslogicaxiomat0000stol/page/5 5] |isbn=9780716704577 |url= https://archive.org/details/setslogicaxiomat0000stol |url-access=registration}}</ref> This property is called [[Axiom of extensionality|extensionality]]. In particular, this implies that there is only one empty set.

Sets are ubiquitous in modern mathematics. Indeed, [[Set theory|set theory]], more specifically [[Zermelo–Fraenkel set theory]], has been the standard way to provide rigorous [[Philosophy:Foundations of mathematics|foundations]] for all branches of mathematics since the first half of the 20th century.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/n11/mode/2up 1]}}

==Definition and notation {{anchor|Definition|Notation}}==
Mathematical texts commonly denote sets by capital letters<ref name="LipschutzLipson1997">{{cite book|author1=Seymor Lipschutz|author2=Marc Lipson|title=Schaum's Outline of Discrete Mathematics|url=https://books.google.com/books?id=6A5g3RiYiBUC&pg=PA1|date=22 June 1997|publisher=McGraw Hill Professional|isbn=978-0-07-136841-4|page=1}}</ref>{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/n11/mode/2up 1]}} in italic, such as {{mvar|A}}, {{mvar|B}}, {{mvar|C}}.<ref name=":1">{{Cite web|title=Introduction to Sets|url=https://www.mathsisfun.com/sets/sets-introduction.html|access-date=2020-08-19|website=www.mathsisfun.com}}</ref> A set may also be called a ''collection'' or ''family'', especially when its elements are themselves sets.

===Roster notation===
'''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:<ref name="Roberts2009">{{cite book|author=Charles Roberts|title=Introduction to Mathematical Proofs: A Transition|url=https://books.google.com/books?id=NjBLnLyE4jAC&pg=PA45|date=24 June 2009|publisher=CRC Press|isbn=978-1-4200-6956-3|page=45}}</ref><ref name="JohnsonJohnson2004">{{cite book|author1=David Johnson|author2=David B. Johnson|author3=Thomas A. Mowry|title=Finite Mathematics: Practical Applications (Docutech Version)|url=https://books.google.com/books?id=ZQAqzxLFXhoC&pg=PA220|date=June 2004|publisher=W. H. Freeman|isbn=978-0-7167-6297-3|page=220}}</ref><ref name="BelloKaul2013">{{cite book|author1=Ignacio Bello|author2=Anton Kaul|author3=Jack R. Britton|title=Topics in Contemporary Mathematics|url=https://books.google.com/books?id=d8Se_8DWTQ4C&pg=PA47|date=29 January 2013|publisher=Cengage Learning|isbn=978-1-133-10742-2|page=47}}</ref><ref name="Epp2010">{{cite book|author=Susanna S. Epp|title=Discrete Mathematics with Applications|url=https://books.google.com/books?id=PPc_2qUhXrAC&pg=PA13|date=4 August 2010|publisher=Cengage Learning|isbn=978-0-495-39132-6|page=13}}</ref>
{{block indent|{{math|1=''A'' = {{mset|4, 2, 1, 3}}}}}}
{{block indent|{{math|1=''B'' = {{mset|blue, white, red}}}}.}}

This notation was introduced by [[Biography:Ernst Zermelo|Ernst Zermelo]] in 1908.<ref>A. Kanamori, "[https://math.bu.edu/people/aki/8.pdf The Empty Set, the Singleton, and the Ordered Pair]", p.278. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</ref> In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a [[Sequence|sequence]], a [[Tuple|tuple]], or a [[Permutation|permutation]] of a set, the ordering of the terms matters). For example, {{math|{{mset|2, 4, 6}}}} and {{math|{{mset|4, 6, 4, 2}}}} represent the same set.<ref name="MaurerRalston2005">{{cite book|author1=Stephen B. Maurer|author2=Anthony Ralston|title=Discrete Algorithmic Mathematics|url=https://books.google.com/books?id=_0vNBQAAQBAJ&pg=PA11|date=21 January 2005|publisher=CRC Press|isbn=978-1-4398-6375-6|page=11}}</ref><ref name=":1" /><ref name="DalenDoets2014">{{cite book|author1=D. Van Dalen|author2=H. C. Doets|author3=H. De Swart|title=Sets: Naïve, Axiomatic and Applied: A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students|url=https://books.google.com/books?id=PfbiBQAAQBAJ&pg=PA1|date=9 May 2014|publisher=Elsevier Science|isbn=978-1-4831-5039-0|page=1}}</ref>

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an [[Ellipsis#In mathematical notation|ellipsis]] '{{math|...}}'.<ref name="BastaDeLong2013">{{cite book|author1=Alfred Basta|author2=Stephan DeLong|author3=Nadine Basta|title=Mathematics for Information Technology|url=https://books.google.com/books?id=VUYLAAAAQBAJ&pg=PA3|date=1 January 2013|publisher=Cengage Learning|isbn=978-1-285-60843-3|page=3}}</ref><ref name="BrackenMiller2013">{{cite book|author1=Laura Bracken|author2=Ed Miller|title=Elementary Algebra|url=https://books.google.com/books?id=nFkrl_kDiTAC&pg=PA36|date=15 February 2013|publisher=Cengage Learning|isbn=978-0-618-95134-5|page=36}}</ref> For instance, the set of the first thousand positive integers may be specified in roster notation as
{{block indent|{{math|{{mset|1, 2, 3, ..., 1000}}}}.}}

====Infinite sets in roster notation====
An [[Infinite set|infinite set]] is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
{{block indent|{{nowrap|{{math|{{mset|0, 1, 2, 3, 4, ...}}}},}}}}
and the set of all [[Integer|integer]]s is
{{block indent|{{nowrap|{{math|{{mset|..., −3, −2, −1, 0, 1, 2, 3, ...}}}}.}}}}

===Semantic definition===
Another way to define a set is to use a rule to determine what the elements are:
{{block indent|Let {{mvar|A}} be the set whose members are the first four positive [[Integer|integers]].}}
{{block indent|Let {{mvar|B}} be the set of colors of the French flag.}}

Such a definition is called a ''semantic description''.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/4/mode/2up 4]}}<ref name="Ruda2011">{{cite book|author=Frank Ruda|title=Hegel's Rabble: An Investigation into Hegel's Philosophy of Right|url=https://books.google.com/books?id=VV0SBwAAQBAJ&pg=PA151|date=6 October 2011|publisher=Bloomsbury Publishing|isbn=978-1-4411-7413-0|page=151}}</ref>

===Set-builder notation===
{{main|Set-builder notation}}

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.<ref name="Ruda2011"/><ref name="Lucas1990">{{cite book|author=John F. Lucas|title=Introduction to Abstract Mathematics|url=https://books.google.com/books?id=jklsb5JUgoQC&pg=PA108|year=1990|publisher=Rowman & Littlefield|isbn=978-0-912675-73-2|page=108}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Set|url=https://mathworld.wolfram.com/Set.html|access-date=2020-08-19|website=Wolfram MathWorld |language=en}}</ref> For example, a set {{mvar|F}} can be defined as follows:

<math display="block">F = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>

In this notation, the [[Vertical bar|vertical bar]] "|" means "such that", and the description can be interpreted as "{{mvar|F}} is the set of all numbers {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.<ref name="Steinlage1987">{{cite book|author=Ralph C. Steinlage|title=College Algebra|url=https://books.google.com/books?id=lcg3gY3444IC|year=1987|publisher=West Publishing Company|isbn=978-0-314-29531-6}}</ref>

===Classifying methods of definition===
[[Philosophy:Philosophy|Philosophy]] uses specific terms to classify types of definitions:
*An ''intensional definition'' uses a ''rule'' to determine membership. Semantic definitions and definitions using set-builder notation are examples.
*An ''extensional definition'' describes a set by ''listing all its elements''.<ref name="Ruda2011"/> Such definitions are also called ''[[Philosophy:Enumerative definition|enumerative]]''.
*An ''[[Philosophy:Ostensive definition|ostensive definition]]'' is one that describes a set by giving ''examples'' of elements; a roster involving an ellipsis would be an example.

==Membership==
{{Main|Element (mathematics)}}
If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''".{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/2/mode/2up 2]}} The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as "''y'' is not in ''B''".<ref name="CapinskiKopp2004">{{cite book|author1=Marek Capinski|author2=Peter E. Kopp|title=Measure, Integral and Probability|url=https://books.google.com/books?id=jdnGYuh58YUC&pg=PA2|year=2004|publisher=Springer Science & Business Media|isbn=978-1-85233-781-0|page=2}}</ref><ref>{{Cite web|title=Set Symbols|url=https://www.mathsisfun.com/sets/symbols.html|access-date=2020-08-19|website=www.mathsisfun.com}}</ref>

For example, with respect to the sets {{math|1=''A'' = {{mset|1, 2, 3, 4}}}}, {{math|1=''B'' = {{mset|blue, white, red}}}}, and {{math|1=''F'' = {{mset|''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19}}}},
{{block indent|{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and}}
{{block indent|{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.}}

==The empty set==
{{main|Empty set}}
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}}, <math>\emptyset</math>, {{mset|&nbsp;}},{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/8/mode/2up 8]}}<ref name="LeungChen1992">{{cite book|author1=K.T. Leung|author2=Doris Lai-chue Chen|title=Elementary Set Theory, Part I/II|url=https://books.google.com/books?id=cdmy2eOhJdkC&pg=PA27|date=1 July 1992|publisher=Hong Kong University Press|isbn=978-962-209-026-2|page=27}}</ref> {{math|ϕ}},<ref>{{cite book|title=Understanding ISC Mathematics Class XI|volume=1|first=M.L.|last=Aggarwal|publisher=Arya Publications (Avichal Publishing Company)|year=2021|chapter=1. Sets|page=A=3}}</ref> or {{mvar|ϕ}}.<ref>{{cite book|title=Chhaya Ganit (Ekadash Shreni)|first=De|last=Sourendra Nath|publisher=Scholar Books Pvt. Ltd.|date=January 2015|chapter=Unit-1 Sets and Functions: 1. Set Theory|page=5}}</ref>

==Singleton sets==
{{main|Singleton (mathematics)}}
A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''.<ref name="Stoll"/> Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
The set {{mset|''x''}} and the element ''x'' mean different things; Halmos{{sfn|Halmos|1960|loc=[https://archive.org/details/naivesettheory00halm/page/4/mode/2up Sect.2]}} draws the analogy that a box containing a hat is not the same as the hat.

==Subsets==
{{Main|Subset}}
If every element of set ''A'' is also in ''B'', then ''A'' is described as being a ''subset of B'', or ''contained in B'', written ''A'' ⊆ ''B'',<ref name="Hausdorff2005">{{cite book|author=Felix Hausdorff|title=Set Theory|url=https://books.google.com/books?id=yvVIdH16k0YC&pg=PA30|year=2005|publisher=American Mathematical Soc.|isbn=978-0-8218-3835-8|page=30}}</ref> or ''B'' ⊇ ''A''.<ref name="Comninos2010">{{cite book|author=Peter Comninos|title=Mathematical and Computer Programming Techniques for Computer Graphics|url=https://books.google.com/books?id=Kdb7-YnnOVwC&pg=PA7|date=6 April 2010|publisher=Springer Science & Business Media|isbn=978-1-84628-292-8|page=7}}</ref> The latter notation may be read ''B contains A'', ''B includes A'', or ''B is a superset of A''. The [[Relation (mathematics)|relationship]] between sets established by ⊆ is called ''inclusion'' or ''containment''. Two sets are equal if they contain each other: ''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' is equivalent to ''A'' = ''B''.<ref name="Lucas1990"/>

If ''A'' is a subset of ''B'', but ''A'' is not equal to ''B'', then ''A'' is called a ''proper subset'' of ''B''. This can be written ''A'' ⊊ ''B''. Likewise, ''B'' ⊋ ''A'' means ''B is a proper superset of A'', i.e. ''B'' contains ''A'', and is not equal to ''A''.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' to mean ''A'' is any subset of ''B'' (and not necessarily a proper subset),{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/2/mode/2up 3]}}<ref name="CapinskiKopp2004"/> while others reserve ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' for cases where ''A'' is a proper subset of ''B''.<ref name="Hausdorff2005"/>

Examples:
* The set of all humans is a proper subset of the set of all mammals.
* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.

The empty set is a subset of every set,{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/8/mode/2up 8]}} and every set is a subset of itself:{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/2/mode/2up 3]}}
* ∅ ⊆ ''A''.
* ''A'' ⊆ ''A''.

==Euler and Venn diagrams==
[[File:Venn A subset B.svg|thumb|150px|''A'' is a subset of ''B''.<br>''B'' is a superset of ''A''.]]
An [[Euler diagram]] is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap. 

A [[Venn diagram]], in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).

==Special sets of numbers in mathematics==
[[File:NumberSetinC.svg|thumb|The natural numbers <math>\mathbb{N}</math> are contained in the integers <math>\mathbb{Z}</math>, which are contained in the rational numbers <math>\mathbb{Q}</math>, which are contained in the real numbers <math>\mathbb{R}</math>, which are contained in the complex numbers <math>\mathbb{C}</math>]]

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. 

Many of these important sets are represented in mathematical texts using bold (e.g. <math>\mathbf  Z</math>) or [[Blackboard bold|blackboard bold]] (e.g. <math>\mathbb Z</math>) typeface.<ref name="Tourlakis2003">{{cite book|author=George Tourlakis|title=Lectures in Logic and Set Theory: Volume 2, Set Theory|url=https://books.google.com/books?id=nparMXao59QC&pg=PA137|date=13 February 2003|publisher=Cambridge University Press|isbn=978-1-139-43943-5|pages=137}}</ref> These include
* <math>\mathbf  N</math> or <math>\mathbb N</math>, the set of all [[Natural number|natural number]]s: <math>\mathbf  N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});<ref name="Tourlakis2003"/>
* <math>\mathbf  Z</math> or <math>\mathbb Z</math>, the set of all [[Integer|integer]]s (whether positive, negative or zero): <math>\mathbf  Z=\{...,-2,-1,0,1,2,3,...\}</math>;<ref name="Tourlakis2003"/>
* <math>\mathbf  Q</math> or <math>\mathbb Q</math>, the set of all [[Rational number|rational number]]s (that is, the set of all proper and improper fractions): <math>\mathbf  Q=\left\{\frac {a}{b}\mid a,b\in\mathbf  Z,b\ne0\right\}</math>. For example, {{math|&minus;{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};<ref name="Tourlakis2003"/>
* <math>\mathbf  R</math> or <math>\mathbb R</math>, the set of all [[Real number|real numbers]], including all rational numbers and all [[Irrational number|irrational]] numbers (which include [[Algebraic number|algebraic number]]s such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as [[Pi|{{mvar|π}}]] and [[E (mathematical constant)|{{math|''e''}}]]);<ref name="Tourlakis2003"/>
* <math>\mathbf  C</math> or <math>\mathbb C</math>, the set of all [[Complex number|complex number]]s: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.<ref name="Tourlakis2003"/>

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. 

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math>\mathbf{Q}^+</math> represents the set of positive rational numbers.

==Functions==
A ''[[Function (mathematics)|function]]'' (or ''[[Map (mathematics)|mapping]]'') from a set {{mvar|A}} to a set {{mvar|B}} is a rule that assigns to each "input" element of {{mvar|A}} an "output" that is an element of {{mvar|B}}; more formally, a function is a special kind of [[Relation (mathematics)|relation]], one that relates each element of {{mvar|A}} to ''exactly one'' element of {{mvar|B}}. A function is called
* injective (or one-to-one) if it maps any two different elements of {{mvar|A}} to ''different'' elements of {{mvar|B}},
* surjective (or onto) if for every element of {{mvar|B}}, there is at least one element of {{mvar|A}} that maps to it, and
* bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of {{mvar|A}} is paired with a unique element of {{mvar|B}}, and each element of {{mvar|B}} is paired with a unique element of {{mvar|A}}, so that there are no unpaired elements.
An injective function is called an ''injection'', a surjective function is called a ''surjection'', and a bijective function is called a ''bijection'' or ''one-to-one correspondence''.

==Cardinality==
{{Main|Cardinality}}

The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}.<ref name="Moschovakis1994">{{cite book|author=Yiannis N. Moschovakis|title=Notes on Set Theory|url=https://books.google.com/books?id=ndx0_6VCypcC|year=1994|publisher=Springer Science & Business Media|isbn=978-3-540-94180-4}}</ref> For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted,<ref name="Fleck2001">{{cite book|author=Arthur Charles Fleck|title=Formal Models of Computation: The Ultimate Limits of Computing|url=https://books.google.com/books?id=c42oYf4zBzMC&pg=PA3|year=2001|publisher=World Scientific|isbn=978-981-02-4500-9|page=3}}</ref><ref name="Johnston2015">{{cite book|author=William Johnston|title=The Lebesgue Integral for Undergraduates|url=https://books.google.com/books?id=v4ueCgAAQBAJ&pg=PA7|date=25 September 2015|publisher=The Mathematical Association of America|isbn=978-1-939512-07-9|page=7}}</ref> so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.

More formally, two sets share the same cardinality if there exists a bijection between them.

The cardinality of the empty set is zero.<ref name="Smith2008">{{cite book|author=Karl J. Smith|title=Mathematics: Its Power and Utility|url=https://books.google.com/books?id=-0x2JszrkooC&pg=PA401|date=7 January 2008|publisher=Cengage Learning|isbn=978-0-495-38913-2|page=401}}</ref>

===Infinite sets and infinite cardinality===
The list of elements of some sets is endless, or ''[[Infinite set|infinite]]''. For example, the set <math>\N</math> of [[Natural number|natural numbers]] is infinite.<ref name="Lucas1990"/> In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have ''infinite cardinality''. 

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of [[Real number|real numbers]] has greater cardinality than the set of natural numbers.<ref name="Stillwell2013">{{cite book|author=John Stillwell|title=The Real Numbers: An Introduction to Set Theory and Analysis|url=https://books.google.com/books?id=VPe8BAAAQBAJ|date=16 October 2013|publisher=Springer Science & Business Media|isbn=978-3-319-01577-4}}</ref> Sets with cardinality less than or equal to that of <math>\N</math> are called ''[[Countable set|countable sets]]''; these are either finite sets or ''countably infinite sets'' (sets of the same cardinality as <math>\N</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <math>\N</math> are called ''[[Uncountable set|uncountable sets]]''.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any [[Line segment|segment]] of that line, of the entire [[Plane (mathematics)|plane]], and indeed of any finite-dimensional [[Euclidean space]].<ref name="Tall2006">{{cite book|author=David Tall|title=Advanced Mathematical Thinking|url=https://books.google.com/books?id=czKqBgAAQBAJ&pg=PA212|date=11 April 2006|publisher=Springer Science & Business Media|isbn=978-0-306-47203-9|pages=211}}</ref>

===The continuum hypothesis===
{{Main|Continuum hypothesis}}
The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line.<ref name = "Cantor1878">{{Cite journal
 | first = Georg | last = Cantor
 | title = Ein Beitrag zur Mannigfaltigkeitslehre
 | journal = Journal für die Reine und Angewandte Mathematik
 | volume = 1878 | issue = 84
 | year = 1878 | pages=242–258
 | url = http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15 | doi=10.1515/crll.1878.84.242}}</ref> In 1963, [[Biography:Paul Cohen|Paul Cohen]] proved that the continuum hypothesis is [[Independence (mathematical logic)|independent]] of the axiom system ZFC consisting of [[Zermelo–Fraenkel set theory]] with the [[Axiom of choice|axiom of choice]].<ref name = "Cohen1963">
{{Cite journal
 | first = Paul J. | last = Cohen
 | title = The Independence of the Continuum Hypothesis
 | journal = Proceedings of the National Academy of Sciences of the United States of America
 | volume = 50 | issue = 6 | date = December 15, 1963 | pages = 1143–1148
 | doi = 10.1073/pnas.50.6.1143
 | pmid = 16578557
 | pmc = 221287 | jstor=71858
 | bibcode = 1963PNAS...50.1143C| doi-access = free
 }}
</ref> (ZFC is the most widely-studied version of axiomatic set theory.)

==Power sets==
{{Main|Power set}}
The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}.<ref name="Lucas1990" /> The [[Empty set|empty set]] and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.<ref name="Lucas1990" />{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/18/mode/2up 19]}}<ref name=":1" />

If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/20/mode/2up 20]}} For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.

If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a [[Bijection|bijection]] from {{math|''S''}} onto {{math|''P''(''S'')}}.)<ref name="BurgerStarbird2004">{{cite book|author1=Edward B. Burger|author2=Michael Starbird|title=The Heart of Mathematics: An invitation to effective thinking|url=https://books.google.com/books?id=M-qK8anbZmwC&pg=PA183|date=18 August 2004|publisher=Springer Science & Business Media|isbn=978-1-931914-41-3|page=183}}</ref>

==Partitions==
{{main|Partition of a set}}

A [[Partition of a set|partition of a set]] ''S'' is a set of nonempty subsets of ''S'', such that every element ''x'' in ''S'' is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the [[Union (set theory)|union]] of all the subsets of the partition is ''S''.<ref name="Mansour2012">{{cite book|author=Toufik Mansour|title=Combinatorics of Set Partitions|url=https://books.google.com/books?id=5NvrH4w8WGsC|date=27 July 2012|publisher=CRC Press|isbn=978-1-4398-6333-6}}</ref>{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/28/mode/2up 28]}}

== Basic operations ==
{{Main|Algebra of sets}}

[[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]]
Suppose that a [[Universe (mathematics)|universal set]] {{mvar|U}} (a set containing all elements being discussed) has been fixed, and that {{mvar|A}} is a subset of {{mvar|U}}.
* The [[Complement (set theory)|complement]] of {{mvar|A}} is the set of all elements (of {{mvar|U}}) that do ''not'' belong to {{mvar|A}}.  It may be denoted {{math|''A''{{sup|c}}}} or {{math|''A''′}}.  In set-builder notation, <math>A^{\text{c}} = \{a \in U : a \notin A \}</math>.  The complement may also be called the ''absolute complement'' to distinguish it from the relative complement below.  Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers.

[[File:Venn0111.svg|thumb|<div class="center">The '''union''' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}</div>]]
[[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of {{mvar|A}} and {{mvar|B}}, denoted {{math|''A'' ∩ ''B''}}</div>]]
[[File:Venn0100.svg|thumb|<div class="center">The '''set difference''' {{math|''A'' \ ''B''}}</div>]]
[[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
Given any two sets {{mvar|A}} and {{mvar|B}}, 
* their [[Union (set theory)|union]] {{math|''A'' ∪ ''B''}} is the set of all things that are members of ''A'' or ''B'' or both.
* their [[Intersection (set theory)|intersection]] {{math|''A'' ∩ ''B''}} is the set of all things that are members of both ''A'' and ''B''.  If {{math|1=''A'' ∩ ''B'' = ∅}}, then {{mvar|A}} and {{mvar|B}} are said to be ''disjoint''.
* the set difference {{math|''A'' \ ''B''}} (also written {{math|''A'' − ''B''}}) is the set of all things that belong to {{mvar|A}} but not {{mvar|B}}. Especially when {{mvar|B}} is a subset of {{mvar|A}}, it is also called the [[Complement (set theory)|relative complement]] of {{mvar|B}} in {{mvar|A}}. With {{math|''B''{{sup|c}}}} as the absolute complement of ''B'' (in the universal set {{mvar|U}}), {{math|1=''A'' \ ''B'' = ''A'' ∩ ''B''{{sup|c}}}} .
* their [[Symmetric difference|symmetric difference]] {{math|''A'' Δ ''B''}} is the set of all things that belong to {{mvar|A}} or {{mvar|B}} but not both.  One has <math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)</math>.
* their [[Cartesian product|cartesian product]] {{math|''A'' × ''B''}} is the set of all ordered pairs {{math|(''a'',''b'')}} such that {{mvar|a}} is an element of {{mvar|A}} and {{mvar|b}} is an element of {{mvar|B}}.

Examples:
* {{math|1={1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}}}.
* {{math|1={1, 2, 3} ∩ {3, 4, 5} = {3}}}.
* {{math|1={1, 2, 3} − {3, 4, 5} = {1, 2}}}.
* {{math|1={1, 2, 3} Δ {3, 4, 5} = {1, 2, 4, 5}}}.
* {{math|1={''a'', ''b''} × {1, 2, 3} = {(''a'',1), (''a'',2), (''a'',3), (''b'',1), (''b'',2), (''b'',3)}}}.

The operations above satisfy many identities.  For example, one of [[De Morgan's laws]] states that '''{{math|(''A'' ∪ ''B'')′ {{=}} ''A''′ ∩ ''B''′}}''' (that is, the elements outside the union of {{mvar|A}} and {{mvar|B}} are the elements that are outside {{mvar|A}} ''and'' outside {{mvar|B}}).

The cardinality of {{math|''A'' × ''B''}} is the product of the cardinalities of {{mvar|A}} and {{mvar|B}}.
(This is an elementary fact when {{mvar|A}} and {{mvar|B}} are finite.  When one or both are infinite, multiplication of cardinal numbers is defined to make this true.)

The power set of any set becomes a [[Boolean ring]] with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.

==Applications==
Sets are ubiquitous in modern mathematics. For example, [[Algebraic structure|structures]] in [[Abstract algebra|abstract algebra]], such as [[Group (mathematics)|groups]], [[Field (mathematics)|fields]] and [[Ring (mathematics)|rings]], are sets [[Closure (mathematics)|closed]] under one or more operations.

One of the main applications of naive set theory is in the construction of [[Relation (mathematics)|relations]]. A relation from a [[Domain of a function|domain]] {{math|''A''}} to a [[Codomain|codomain]] {{math|''B''}} is a subset of the Cartesian product {{math|''A'' × ''B''}}. For example, considering the set {{math|''S'' {{=}} {{mset|rock, paper, scissors}}}} of shapes in the [[Rock paper scissors|game]] of the same name, the relation "beats" from {{math|''S''}} to {{math|''S''}} is the set {{math|''B'' {{=}} {{mset|(scissors,paper), (paper,rock), (rock,scissors)}}}}; thus {{math|''x''}} beats {{math|''y''}} in the game if the pair {{math|(''x'',''y'')}} is a member of {{math|''B''}}. Another example is the set {{math|''F''}} of all pairs {{math|(''x'', ''x''{{sup|2}})}}, where {{math|''x''}} is real. This relation is a subset of {{math|'''R''' × '''R'''}}, because the set of all squares is subset of the set of all real numbers. Since for every {{math|''x''}} in {{math|'''R'''}}, one and only one pair {{math|(''x'',...)}} is found in {{math|''F''}}, it is called a [[Function (mathematics)|function]]. In functional notation, this relation can be written as {{math|''F''(''x'') {{=}} ''x''{{sup|2}}}}.
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==Principle of inclusion and exclusion==
{{Main|Inclusion–exclusion principle}}
[[Image:A union B.svg|thumb|The inclusion-exclusion principle for two finite sets states that the size of their union is the sum of the sizes of the sets minus the size of their intersection.]]

The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically as
<math display="block"> |A \cup B| = |A| + |B| - |A \cap B|.</math>

A more general form of the principle gives the cardinality of any finite union of finite sets:
<math display="block">\begin{align}
\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|=& \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right) \\
&{} - \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right) \\
&{} + \ldots \\
&{} + \left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right).
\end{align}</math>


==History==
{{Main|Set theory}}
The concept of a set emerged in mathematics at the end of the 19th century.<ref name="Ferreirós2007">{{cite book|author=José Ferreirós|title=Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics|url=https://books.google.com/books?id=TXRBwwEACAAJ|date=16 August 2007|publisher=Birkhäuser Basel|isbn=978-3-7643-8349-7}}</ref> The German word for set, ''Menge'', was coined by [[Biography:Bernard Bolzano|Bernard Bolzano]] in his work ''[[Paradoxes of the Infinite]]''.<ref name="Russ2004">{{cite book|author=Steve Russ|title=The Mathematical Works of Bernard Bolzano|url=https://books.google.com/books?id=zp7cLQn0x3gC&pg=PR28|date=9 December 2004|publisher=OUP Oxford|isbn=978-0-19-151370-1}}</ref><ref name="EwaldEwald1996">{{cite book|author1=William Ewald|author2=William Bragg Ewald|title=From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics|url=https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA249|year=1996|publisher=OUP Oxford|isbn=978-0-19-850535-8|page=249}}</ref><ref name="RusnockSebestík2019">{{cite book|author1=Paul Rusnock|author2=Jan Sebestík|title=Bernard Bolzano: His Life and Work|url=https://books.google.com/books?id=-hqJDwAAQBAJ&pg=PA430|date=25 April 2019|publisher=OUP Oxford|isbn=978-0-19-255683-7|page=430}}</ref>[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage with a translation of the original set definition of Georg Cantor. The German word ''Menge'' for ''set'' is translated with ''aggregate'' here.]] [[Biography:Georg Cantor|Georg Cantor]], one of the founders of set theory,   gave the following definition at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':<ref name="Cantor.1895">{{cite journal | url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00225557X | author=Georg Cantor | title=Beiträge zur Begründung der transfiniten Mengenlehre (1) | journal=Mathematische Annalen | volume=46 | number=4 | pages=481&ndash;512 | date=Nov 1895 |language=German}}</ref><ref name="Cantor">{{cite book |quote=By an 'aggregate' (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) ''M'' of definite and separate objects ''m'' of our intuition or our thought.|last1=Cantor |first1=Georg |last2=Jourdain |first2=((Philip E.B. (Translator))) |date=1915  |title=Contributions to the founding of the theory of transfinite numbers |publisher=New York Dover Publications (1954 English translation) }} Here: p.85</ref>

{{Blockquote|A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.}}

[[Biography:Bertrand Russell|Bertrand Russell]] introduced the distinction between a set and a class (a set is a class, but some classes, such as the class of all sets, are not sets; see [[Russell's paradox]]):<ref>[[Biography:Bertrand Russell|Bertrand Russell]] (1903) ''[[The Principles of Mathematics]]'', chapter VI: Classes</ref> 

{{Blockquote|When mathematicians deal with what they call a manifold, aggregate, ''Menge'', ''ensemble'', or some equivalent name, it is  common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case ''is'' the class.}}

===Naive set theory===
{{Main|Naive set theory}}
The foremost property of a set is that it can have elements, also called ''members''. Two sets are [[Equality (mathematics)|equal]] when they have the same elements. More precisely, sets ''A'' and ''B'' are equal if every element of ''A'' is an element of ''B'', and every element of ''B'' is an element of ''A''; this property is called the ''[[Extensionality#In mathematics|extensionality]] of sets''.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/2/mode/2up 2]}} 

The simple concept of a set has proved enormously useful in mathematics, but [[:Category:Paradoxes of naive set theory|paradoxes]] arise if no restrictions are placed on how sets can be constructed:
* [[Russell's paradox]] shows that the "set of all sets that ''do not contain themselves''", i.e., {{mset|''x'' | ''x'' is a set and ''x'' ∉ ''x''}}, cannot exist.
* [[Cantor's paradox]] shows that "the set of all sets" cannot exist.

[[Naive set theory|Naïve set theory]] defines a set as any ''[[Well-defined|well-defined]]'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.

===Axiomatic set theory===
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a [[Primitive notion|primitive notion]].<ref name="Ferreiros2001">{{cite book|author=Jose Ferreiros|title=Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics|url=https://books.google.com/books?id=DITy0nsYQQoC|date=1 November 2001|publisher=Springer Science & Business Media|isbn=978-3-7643-5749-8}}</ref> The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using [[First-order logic|first-order logic]]. According to [[Gödel's incompleteness theorems]] however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.{{cn|date=February 2021}}

==See also==
{{Div col|colwidth=20em}}

* [[Algebra of sets]]
* [[Alternative set theory]]
* [[Category of sets]]
* [[Class (set theory)]]
* [[Dense set]]
* [[Family of sets]]
* [[Fuzzy set]]
* [[Internal set]]
* [[Mereology]]
* [[Multiset]]
* [[Principia Mathematica]]
* [[Rough set]]
{{Div col end}}

==Notes==
{{Reflist}}

==References==
*{{cite book |last=Dauben |first=Joseph W. |title=Georg Cantor: His Mathematics and Philosophy of the Infinite |location=Boston |publisher=Harvard University Press |year=1979 |isbn=0-691-02447-2 |url-access=registration |url=https://archive.org/details/georgcantorhisma0000daub }}
*{{cite book |last=Halmos |first=Paul R. |title=Naive Set Theory |url=https://archive.org/details/naivesettheory00halm |url-access=registration |location=Princeton, N.J. |publisher=Van Nostrand |year=1960 |isbn=0-387-90092-6 }}
*{{cite book |last=Stoll |first=Robert R. |title=Set Theory and Logic |location=Mineola, N.Y. |publisher=[[Social:Dover Publications|Dover Publications]] |year=1979 |isbn=0-486-63829-4 }}
*{{cite book |last=Velleman |first=Daniel |title=How To Prove It: A Structured Approach |publisher={{wipe|Cambridge University Press}} |year=2006 |isbn=0-521-67599-5 }}

==External links==
*[https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0046/LOG_0044.pdf Cantor's "Beiträge zur Begründung der transfiniten Mengenlehre"] {{in lang|de}}

{{Mathematical logic}}

[[Category:Concepts in logic]]
[[Category:Set theory|*]]

{{Sourceattribution|Set (mathematics)}}