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Reorganize properties section. The modeling of zero as the empty set doesn't really belong here. |
Undid revision 1226065701 by PBUK (talk) this strikes me as a worse way to convey the relevant information |
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{{Other uses of|Empty}}
[[File:Nullset.svg|thumb|upright=0.
In [[mathematics]], the '''empty set''' or '''void set''' is the unique [[Set (mathematics)|set]] having no [[Element (mathematics)|elements]]; its size or [[cardinality]] (count of elements in a set) is [[0|zero]].<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Empty Set|url=https://mathworld.wolfram.com/EmptySet.html|access-date=2020-08-11|website=mathworld.wolfram.com|language=en}}</ref> Some [[axiomatic set theories]] ensure that the empty set exists by including an [[axiom of empty set]], while in other theories, its existence can be deduced. Many possible properties of sets are [[vacuously true]] for the empty set.
Any set other than the empty set is called
In some textbooks and popularizations, the empty set is referred to as the "null set".<ref name=":1" /> However,
==Notation==
{{Main|Null sign}}
[[Image:Empty set symbol.svg|thumb|upright=0.
Common notations for the empty set include "{ }", "<math>\emptyset</math>", and "∅". The latter two symbols were introduced by the [[Bourbaki group]] (specifically [[André Weil]]) in 1939, inspired by the letter [[Ø]] ({{unichar|d8|LATIN CAPITAL LETTER O WITH STROKE}}) in the [[Danish orthography|Danish]] and [[Norwegian orthography|Norwegian]] alphabets.<ref>{{cite web| url = http://jeff560.tripod.com/set.html| title = Earliest Uses of Symbols of Set Theory and Logic.}}</ref> In the past, "0" (the numeral [[zero]]) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.<ref>{{Cite book|url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=007054235X|edition=3rd|pages=300}}</ref>
The symbol ∅ is available at [[Unicode]] point
When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.<ref>e.g. Nina Grønnum (2005, 2013) ''Fonetik og Fonologi: Almen og dansk.'' Akademisk forlag, Copenhagen.</ref>
== Properties ==
In standard [[axiomatic set theory]], by the [[axiom of extensionality|principle of extensionality]], two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".
The only subset of the empty set is the empty set itself; equivalently, the [[power set]] of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its [[cardinality]]) is zero. The empty set is the only set with either of these properties.
* The [[power set]] of the empty set is the set containing only the empty set:▼
[[For any]] set ''A'':
* The empty set is a [[subset]] of ''A''
* The [[
* The [[
▲* The [[
▲* The [[intersection (set theory)|intersection]] of ''A'' with the empty set is the empty set:
For any [[property (philosophy)|property]] ''P'':
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=== Operations on the empty set ===
When speaking of the [[summation|sum]] of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the [[empty sum]]) is zero. The reason for this is that zero is the [[identity element]] for addition. Similarly, the [[multiplication|product]] of the elements of the empty set (the [[empty product]]) should be considered to be [[1 (number)|one]]
A [[derangement]] is a [[permutation]] of a set without [[fixed point (mathematics)|fixed point]]s. The empty set can be considered a derangement of itself, because it has only one permutation (<math>0!=1</math>), and it is vacuously true that no element (of the empty set) can be found that retains its original position.
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==Questioned existence==
=== Historical issues ===
In the context of sets of real numbers, Cantor used <math>P\equiv O</math> to denote "<math>P</math> contains no single point". This <math>\equiv O</math> notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed <math>O</math> as an existent set on its own, or if Cantor merely used <math>\equiv O</math> as an emptiness predicate. Zermelo accepted <math>O</math> itself as a set, but considered it an "improper set".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/8.pdf The Empty Set, the Singleton, and the Ordered Pair]", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</ref>
=== Axiomatic set theory ===
In [[Zermelo set theory]], the existence of the empty set is assured by the [[axiom of empty set]], and its uniqueness follows from the [[axiom of extensionality]]. However, the axiom of empty set can be shown redundant in at least two ways:
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is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is <math>\varnothing</math>" and the latter to "The set {ham sandwich} is better than the set <math>\varnothing</math>". The first compares elements of sets, while the second compares the sets themselves.<ref name="Darling">{{cite book|title=The Universal Book of Mathematics|author=D. J. Darling|publisher= [[John Wiley and Sons]]|year=2004 |isbn=0-471-27047-4|page=106}}</ref>
[[E. J. Lowe (philosopher)|Jonathan Lowe]] argues that while the empty set
:
it is also the case that:
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==Further reading==
*[[Paul Halmos|Halmos, Paul]], ''[[Naive Set Theory (book)|Naive Set Theory]]''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. {{ISBN|0-387-90092-6}} (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. {{ISBN|978-1-61427-131-4}} (paperback edition).
*{{
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== External links ==
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{{DEFAULTSORT:Empty Set}}
[[Category:Basic concepts in set theory]]
[[Category:0 (number)]]
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