Empty set: Difference between revisions
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{{Other uses of|Empty}} | {{Other uses of|Empty}} | ||
[[File:Nullset.svg|thumb|upright=0.6|The empty set is the set containing no elements.]] | [[File:Nullset.svg|thumb|upright=0.6|alt={}|The empty set is the set containing no elements.|class=skin-invert-image]] | ||
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. | 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. | ||
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==Notation== | ==Notation== | ||
{{Main|Null sign}} | {{Main|Null sign}} | ||
[[Image:Empty set symbol.svg|thumb|upright=0.6|A symbol for the empty set]] | [[Image:Empty set symbol.svg|thumb|upright=0.6|alt=∅|A symbol for the empty set|class=skin-invert-image]] | ||
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> | 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> | ||
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A topological space <math>X</math> is said to have the [[indiscrete topology]] if the only open sets are <math>\varnothing</math> and the entire space. | A topological space <math>X</math> is said to have the [[indiscrete topology]] if the only open sets are <math>\varnothing</math> and the entire space. | ||
The [[Closure (mathematics)|closure]] of the empty set is empty. This is known as "preservation of [[nullary]] [[Union (set theory)|unions]]".<ref>{{cite book |last1=Munkres |first1=James Raymond |title=Topology |date=2018 |publisher=Pearson |location=New York, NY |isbn=978-0134689517 |edition=Second, reissue}}</ref> | The [[Closure (mathematics)|closure]] of the empty set is empty. This is known as "preservation of [[nullary]] [[Union (set theory)|unions]]".<ref>{{cite book |last1=Munkres |first1=James Raymond |author-link=James Munkres |title=Topology |date=2018 |publisher=Pearson |location=New York, NY |isbn=978-0134689517 |edition=Second, reissue}}</ref> | ||
=== Category theory === | === Category theory === | ||
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* {{annotated link|Nothing}} | * {{annotated link|Nothing}} | ||
* {{annotated link|Power set}} | * {{annotated link|Power set}} | ||
* {{annotated link|Zero (linguistics)}} | |||
==References== | ==References== | ||