Editing Set (mathematics) (section)

===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>