Editing Set (mathematics) (section)

==Basic notions== 

In mathematics, a set is a collection of different things.<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><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> These things are called ''elements'' or ''members'' of the set and are typically [[mathematical object]]s of any kind such as numbers, symbols, [[Point (geometry)|points in space]], [[Line (geometry)|lines]], other [[Shape|geometrical shapes]], [[Variable (mathematics)|variables]], [[Function (mathematics)|functions]], or even other sets.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/n11/mode/2up 1]}}<ref>{{Cite book |last=Maddocks |first=J. R. |title=The Gale Encyclopedia of Science |publisher=Gale |year=2004 |isbn=0-7876-7559-8 |editor-last=Lerner |editor-first=K. Lee |pages=3587–3589 |language=en |editor-last2=Lerner |editor-first2=Brenda Wilmoth}}</ref> A set may also be called a ''collection'' or family, especially when its elements are themselves sets; this may avoid the confusion between the set and its members, and may make reading easier. A set may be specified either by listing its elements or by a property that characterizes its elements, such as for the set of the [[Prime number|prime numbers]] or the set of all students in a given class.<ref name=":0">{{Cite book |last=Devlin |first=Keith J. |title=Sets, Functions and Logic: Basic concepts of university mathematics |publisher=Springer |year=1981 |isbn=978-0-412-22660-1 |pages= |language=en |chapter=Sets and functions}}</ref><ref>{{Cite web |title=Set - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Set |access-date=2025-02-06 |website=encyclopediaofmath.org}}</ref><ref>{{Cite web |last=Publishers |first=HarperCollins |title=The American Heritage Dictionary entry: set |url=https://www.ahdictionary.com/word/search.html?q=set |access-date=2025-02-06 |website=www.ahdictionary.com}}</ref>

If {{tmath|x}} is an element of a set {{tmath|S}}, one says that {{tmath|x}} ''belongs'' to {{tmath|S}} or ''is in'' {{tmath|S}}, and this is written as {{tmath|x\in S}}.{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/2/mode/2up 2]}} The statement "{{tmath|y}} is not in {{tmath|S\,}}" is written as {{tmath|y\not\in S}}, 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, if {{tmath|\Z}} is the set of the [[integer]]s, one has {{tmath|-3\in\Z}} and {{tmath|1.5 \not\in \Z}}. Each set is uniquely characterized by its elements. In particular, 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, called [[axiom of extensionality|extensionality]], can be written in formula as
<math display="block">A=B \iff \forall x\; (x\in A \iff x \in B).</math>This implies that there is only one set with no element, the ''[[empty set]]'' (or ''null set'') that is denoted {{tmath|\varnothing, \empty}},{{efn|Some typographical variants are occasionally used, such as {{math|ϕ}},<ref>{{cite book |last=Aggarwal |first=M.L. |title=Understanding ISC Mathematics Class XI |publisher=Arya Publications (Avichal Publishing Company) |year=2021 |volume=1 |page=A=3 |chapter=1. Sets}}</ref> or {{mvar|ϕ}}.<ref>{{cite book |last=Sourendra Nath |first=De |title=Chhaya Ganit (Ekadash Shreni) |date=January 2015 |publisher=Scholar Books Pvt. Ltd. |page=5 |chapter=Unit-1 Sets and Functions: 1. Set Theory}}</ref>}} or {{tmath|\{\,\}.}}{{sfn|Halmos|1960|p=[https://archive.org/details/naivesettheory00halm/page/8/mode/2up 8]}}<ref name="LeungChen1992">{{cite book |author1=K.T. Leung |url=https://books.google.com/books?id=cdmy2eOhJdkC&pg=PA27 |title=Elementary Set Theory, Part I/II |author2=Doris Lai-chue Chen |date=1 July 1992 |publisher=Hong Kong University Press |isbn=978-962-209-026-2 |page=27}}</ref> A ''[[singleton (mathematics)|singleton]]'' is a set with exactly one element.{{efn|The term ''unit set'' is also occasionally used.<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>}} If {{tmath|x}} is this element, the singleton is denoted {{tmath|\{x\}.}} If {{tmath|x}} is itself a set, it must not be confused with {{tmath|\{x\}.}} For example, {{tmath|\empty}} is a set with no elements, while {{tmath|\{\empty\} }} is a singleton with {{tmath|\empty}} as its unique element.

A set is ''[[finite set|finite]]'' if there exists a [[natural number]] {{tmath|n}} such that the {{tmath|n}} first natural numbers  can be put in [[one to one correspondence]]  with the elements of the set. In this case, one says that {{tmath|n}} is the number of elements of the set. A set is ''[[infinite set|infinite]]'' if such an {{tmath|n}} does not exist. The ''empty set'' is a finite set with {{tmath|0}} elements. 

[[File:number-systems.svg|thumb|All standard number systems are infinite sets]]
The natural numbers form an infinite set, commonly denoted {{tmath|\N}}. Other examples of infinite sets include [[number set]]s that contain the natural numbers, [[real vector space]]s, [[curve (mathematics)|curve]]s and most sorts of [[space (mathematics)|space]]s.