Editing Set (mathematics) (section)

== Specifying a set ==
Extensionality implies that for specifying a set, one has either to list its elements or to provide a property that uniquely characterizes the set elements.  

===Roster notation===
''Roster'' or ''enumeration notation'' is a notation introduced by [[Ernst Zermelo]] in 1908 that specifies a set by listing its elements between [[curly bracket|braces]], separated by commas.<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><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|first=David |last=Johnson|first2=David B. |last2=Johnson|first3=Thomas A. |last3=Mowry|title=Finite Mathematics: Practical Applications |edition=Docutech|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|first=Ignacio |last=Bello|first2=Anton |last2=Kaul|first3=Jack R. |last3=Britton|title=Topics in Contemporary Mathematics|url=https://books.google.com/books?id=d8Se_8DWTQ4C&pg=PA47|date=29 January 2013|publisher=Cengage |isbn=978-1-133-10742-2|page=47}}</ref><ref name="Epp2010">{{cite book|first=Susanna S. |last=Epp|title=Discrete Mathematics with Applications|url=https://books.google.com/books?id=PPc_2qUhXrAC&pg=PA13|date=4 August 2010|publisher=Cengage |isbn=978-0-495-39132-6|page=13}}</ref> For example, one knows that 
<math>\{4, 2, 1, 3\}</math>
and
<math>\{\text{blue, white, red}\}</math>
denote sets and not [[tuples]] because of the enclosing braces.

Above notations {{tmath|\{\,\} }} and {{tmath|\{x\} }} for the empty set and for a singleton are examples of roster notation.

When specifying sets, it only matters whether each distinct element is in the set or not; this means a set does not change if elements are repeated or arranged in a different order. For example,<ref>{{cite book|first=Stephen B. |last=Maurer|first2=Anthony |last2=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">{{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><ref name="DalenDoets2014">{{cite book|first=D. |last=Van Dalen|first2=H. C. |last2=Doets|first3=H. |last3=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>

<math display =block>\{1,2,3,4\}=\{4, 2, 1, 3\} = \{4, 2, 4, 3, 1, 3\}.</math>

When there is a clear pattern for generating all set elements, one can use [[Ellipsis#In mathematical notation|ellipses]] for abbreviating the notation,<ref name="BastaDeLong2013">{{cite book|first=Alfred |last=Basta|first2=Stephan |last2=DeLong|first3=Nadine |last3=Basta|title=Mathematics for Information Technology|url=https://books.google.com/books?id=VUYLAAAAQBAJ&pg=PA3|date=1 January 2013|publisher=Cengage |isbn=978-1-285-60843-3|page=3}}</ref><ref name="BrackenMiller2013">{{cite book|first=Laura |last=Bracken|first2=Ed |last2=Miller|title=Elementary Algebra|url=https://books.google.com/books?id=nFkrl_kDiTAC&pg=PA36|date=15 February 2013|publisher=Cengage |isbn=978-0-618-95134-5|page=36}}</ref> such as in 
<math display =block>\{1,2,3,\ldots,1000\}</math>
for the positive integers not greater than {{tmath|1000}}.

Ellipses allow also expanding roster notation to some infinite sets. For example, the set of all integers can be denoted as

<math display =block>\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}</math>

or

<math display =block>\{0, 1, -1, 2, -2, 3, -3, \ldots\}.</math>

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

Set-builder notation specifies a set as being the set of all elements that satisfy some [[logical formula]].<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><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> More precisely, if {{tmath|P(x)}} is a logical formula depending on a [[variable (mathematics)|variable]] {{tmath|x}}, which evaluates to ''true'' or ''false'' depending on the value of {{tmath|x}}, then 
<math display=block>\{x \mid P(x)\}</math>
or<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>
<math display=block>\{x : P(x)\}</math>
denotes the set of all {{tmath|x}} for which {{tmath|P(x)}} is true.<ref name=":0" /> For example, a set {{mvar|F}} can be specified 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]] "|" is read as "such that", and the whole formula can be read as "{{mvar|F}} is the set of all  {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive".

Some logical formulas, such as {{tmath| \color{red}{S \text{ is a set} } }} or {{tmath|\color{red}{S \text{ is a set and } S\not\in S} }} cannot be used in set-builder notation because there is no set for which the elements are characterized by the formula. There are several ways for avoiding the problem. One may prove that the formula defines a set; this is often almost immediate, but may be very difficult. 

One may also introduce a larger set {{tmath|U}} that must contain all elements of the specified set, and write the notation as
<math display=block>\{x\mid x\in U \text{ and ...}\}</math>
or
<math display=block>\{x\in U\mid \text{ ...}\}.</math>

One may also define {{tmath|U}} once for all and take the convention that every variable that appears on the left of the vertical bar of the notation represents an element of {{tmath|U}}. This amounts to say that {{tmath|x\in U}} is implicit in set-builder notation. In this case, {{tmath|U}} is often called ''the [[domain of discourse]]'' or a ''[[universe (mathematics)|universe]]''.

For example, with the convention that a lower case Latin letter may represent a [[real number]] and nothing else, the [[expression (mathematics)|expression]]
<math display=block>\{x\mid x\not\in \Q\}</math>
is an abbreviation of 
<math display="block">\{x\in \R \mid x\not\in \Q\},</math>
which defines the [[irrational number]]s.