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| {{redirect|∅|similar symbols|Ø (disambiguation)}}
| | Hello! I am Lorna. I am satisfied that I could unite to the whole globe. I live in Australia, in the VIC region. I dream to go to the various nations, to get acquainted with appealing individuals.<br><br>My blog post - [http://www.aireys.co.nz/cheapghd.html ghd nz] |
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| [[File:Nullset.svg|thumb|100px|The empty set is the set containing no elements.]]
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| In [[mathematics]], and more specifically [[set theory]], the '''empty set''' is the unique [[Set (mathematics)|set]] having no [[Element (mathematics)|elements]]; its size or [[cardinality]] (count of elements in a set) is [[0 (number)|zero]]. Some [[axiomatic set theories]] assure that the empty set exists by including an [[axiom of empty set]]; in other theories, its existence can be deduced. Many possible properties of sets are [[Trivial (mathematics)|trivially]] true for the empty set.
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| ''[[Null set]]'' was once a common synonym for "empty set", but is now a technical term in [[measure theory]].
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| ==Notation==
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| [[Image:Empty set.svg|thumb|right|100px|A symbol for the empty set]]
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| Common notations for the empty set include "{}", "Ø", and "<math>\emptyset</math>". The latter two symbols were introduced by the [[Bourbaki group]] (specifically [[André Weil]]) in 1939, inspired by the letter [[Ø]] in the [[Danish and Norwegian alphabet]] (and not related in any way to the Greek letter [[phi (letter)|Φ]]).<ref>[http://jeff560.tripod.com/set.html Earliest Uses of Symbols of Set Theory and Logic.]</ref> Other notations for the empty set include "Λ" and "0".<ref>[[John B. Conway]], ''Functions of One Complex Variable'', 2nd ed. P. 12.</ref>
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| The empty-set symbol {{unicode|∅}} is found at [[Unicode]] point U+2205.<ref>[http://www.unicode.org/charts/PDF/U2200.pdf Unicode Standard 5.2]</ref> In [[TeX]], it is coded as <tt>\emptyset</tt> or <tt>\varnothing</tt>.
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| == Properties ==
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| In standard [[axiomatic set theory]], by the [[axiom of extensionality|principle of extensionality]], two sets are equal if they have the same elements; therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of "the empty set" rather than "an empty set".
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| The mathematical symbols employed below are explained [[table of mathematical symbols|here]].
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| [[For any]] set ''A'':
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| * The empty set is a [[subset]] of ''A'':
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| *:<math>\forall A: \varnothing \subseteq A\, .</math>
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| * The [[union (set theory)|union]] of ''A'' with the empty set is ''A'':
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| *:<math>\forall A: A \cup \varnothing = A\, .</math>
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| * The [[intersection (set theory)|intersection]] of ''A'' with the empty set is the empty set:
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| *:<math>\forall A: A \cap \varnothing = \varnothing\, .</math>
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| * The [[Cartesian product]] of ''A'' and the empty set is the empty set:
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| *:<math>\forall A: A \times \varnothing = \varnothing\, .</math>
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| The empty set has the following properties:
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| * Its only subset is the empty set itself:
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| *:<math>\forall A: A \subseteq \varnothing \Rightarrow A = \varnothing\, .</math>
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| * The [[power set]] of the empty set is a set containing only the empty set:
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| *:<math>2^{\varnothing} = \{\varnothing\}\, .</math>
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| * Its number of elements (that is, its [[cardinality]]) is zero:
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| *:<math>| \varnothing | = 0\, .</math>
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| The connection between the empty set and zero goes further, however: in the standard [[set-theoretic definition of natural numbers]], we use sets to [[model theory|model]] the natural numbers. In this context, zero is modelled by the empty set.
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| For any [[property (philosophy)|property]]:
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| * For every element of <math>\varnothing</math> the property holds ([[vacuous truth]]);
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| * There is no element of <math>\varnothing</math> for which the property holds.
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| Conversely, if for some property and some set ''V'', the following two statements hold:
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| * For every element of ''V'' the property holds;
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| * There is no element of ''V'' for which the property holds,
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| :then <math>V = \varnothing</math>.
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| By the definition of [[subset]], the empty set is a subset of any set ''A'', as ''every'' element ''x'' of <math>\varnothing</math> belongs to ''A''. If it is not true that every element of <math>\varnothing</math> is in ''A'', there must be at least one element of <math>\varnothing</math> that is not present in ''A''. Since there are ''no'' elements of <math>\varnothing</math> at all, there is no element of <math>\varnothing</math> that is not in ''A''. Hence every element of <math>\varnothing</math> is in ''A'', and <math>\varnothing</math> is a subset of ''A''. Any statement that begins "for every element of <math>\varnothing</math>" is not making any substantive claim; it is a [[vacuous truth]]. This is often paraphrased as "everything is true of the elements of the empty set."
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| === Operations on the empty set ===
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| Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the [[sum]] of the elements of the empty set is zero, but the [[multiplication|product]] of the elements of the empty set is [[1 (number)|one]] (see [[empty product]]). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the [[identity element]] for addition, and one is the identity element for multiplication.
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| ==In other areas of mathematics==
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| === Extended real numbers ===
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| Since the empty set has no members, when it is considered as a subset of any [[ordered set]], then every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the [[real number line]], every real number is both an upper and lower bound for the empty set.<ref>Bruckner, A.N., Bruckner, J.B., and Thomson, B.S., 2008. ''[http://classicalrealanalysis.com/download.aspx Elementary Real Analysis]'', 2nd ed. Prentice Hall. P. 9.</ref> When considered as a subset of the [[extended reals]] formed by adding two "numbers" or "points" to the real numbers, namely [[negative infinity]], denoted <math>-\infty\!\,,</math> which is defined to be less than every other extended real number, and [[positive infinity]], denoted <math>+\infty\!\,,</math> which is defined to be greater than every other extended real number, then:
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| :<math>\sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty,</math>
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| and
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| :<math>\inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty.</math>
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| That is, the least upper bound (sup or [[supremum]]) of the empty set is negative infinity, while the greatest lower bound (inf or [[infimum]]) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for minimum and infimum.
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| ===Topology===
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| Considered as a subset of the [[real number line]] (or more generally any [[topological space]]), the empty set is both [[closed set|closed]] and [[open set|open]]; it is an example of a [[clopen set|"clopen" set]]. All its [[boundary (topology)|boundary points]] (of which there are none) are in the empty set, and the set is therefore closed; while for every one of its points (of which there are again none), there is an [[open neighbourhood]] in the empty set, and the set is therefore open. Moreover, the empty set is a [[compact set]] by the fact that every [[finite set]] is compact.
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| The [[closure (mathematics)|closure]] of the empty set is empty. This is known as "preservation of [[nullary]] [[union (set theory)|unions]]."
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| === Category theory ===
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| If ''A'' is a set, then there exists precisely one [[function (mathematics)|function]] ''f'' from {} to ''A'', the [[empty function]]. As a result, the empty set is the unique [[initial object]] of the [[category theory|category]] of sets and functions.
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| The empty set can be turned into a [[topological space]], called the empty space, in just one way: by defining the empty set to be [[open set|open]]. This empty topological space is the unique initial object in the [[category of topological spaces]] with [[continuous function (topology)|continuous maps]].
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| ==Questioned existence==
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| === Axiomatic set theory ===
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| 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 either of two ways:
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| *There is already an axiom implying the existence of at least one set. Given such an axiom together with the [[axiom of separation]], the existence of the empty set is easily proved.
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| *In the presence of [[urelement]]s, it is easy to prove that at least one set exists, viz. the set of all urelements. Again, given the [[axiom of separation]], the empty set is easily proved.
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| === Philosophical issues ===
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| While the empty set is a standard and widely accepted mathematical concept, it remains an [[ontological]] curiosity, whose meaning and usefulness are debated by philosophers and logicians.
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| The empty set is not the same thing as ''nothing''; rather, it is a set with nothing ''inside'' it and a set is always ''something''. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all [[chess opening|opening moves]] in [[chess]] that involve a [[king (chess)|king]]."<ref name="Darling" />
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| The popular [[syllogism]]
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| :Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
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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>". It is noted that 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>
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| [[Jonathan Lowe]] argues that while the empty set: | |
| :"...was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object."
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| it is also the case that:
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| :"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a ''set'' which has no members. We cannot conjure such an entity into existence by mere stipulation."<ref name="Lowe">{{cite book|title=Locke|author=E. J. Lowe|publisher=Routledge|year=2005|page=87}}</ref>
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| [[George Boolos]] argued that much of what has been heretofore obtained by set theory can just as easily be obtained by [[plural quantification]] over individuals, without [[wikt:reification|reifying]] sets as singular entities having other entities as members.<ref>*[[George Boolos]], 1984, "To be is to be the value of a variable," ''The Journal of Philosophy'' 91: 430–49. Reprinted in his 1998 ''Logic, Logic and Logic'' ([[Richard Jeffrey]], and Burgess, J., eds.) Harvard Univ. Press: 54–72.</ref>
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| ==See also==
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| *[[Inhabited set]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *[[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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| *{{Citation|last=Jech|first=Thomas|authorlink=Thomas Jech|year=2002|title=Set Theory|edition=3rd millennium|series=Springer Monographs in Mathematics|publisher=Springer|isbn=3-540-44085-2}}
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| == External links ==
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| *{{MathWorld |title=Empty Set |id=EmptySet }}
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| {{logic}}
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| {{Set theory}}
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| {{DEFAULTSORT:Empty Set}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Nothing]]
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| [[Category:Zero]]
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