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| {{redirect|Equipollence|the concept in geometry|Equipollence (geometry)}}
| | Jerrie Swoboda is what somebody can call me along with I totally dig that name. What me and my family genuinely like is acting but All of us can't make it all of my profession really. The job I've been taking up for years is a people manager. Guam is where I've always been living. You can sometimes find my website here: http://prometeu.net<br><br>Here is my webpage - [http://prometeu.net clash of clans trainer] |
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| In [[mathematics]], two [[set (mathematics)|set]]s ''A'' and ''B'' are '''equinumerous''' if there exists a [[one-to-one correspondence]] (a bijection) between them, i.e. if there exists a [[function (mathematics)|function]] from ''A'' to ''B'' such that for every [[element (mathematics)|element]] ''y'' of ''B'' there is exactly one element ''x'' of ''A'' with {{nowrap begin}}''f''(''x'') = ''y''{{nowrap end}}.<ref name="suppes"/> This definition can be applied to both [[finite set|finite]] and [[infinite set]]s and allows one to state that two sets have the same size even if they are infinite.
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| The study of cardinality is often called '''equinumerosity''' (''equalness-of-number''). The terms '''equipollence''' (''equalness-of-strength'') and '''equipotence''' (''equalness-of-power'') are sometimes used instead. The statement that two sets ''A'' and ''B'' are equinumerous is usually denoted
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| :<math>A \approx B \,</math> or <math>A \sim B</math>, or <math>|A|=|B|.</math>
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| [[Georg Cantor]], the inventor of [[set theory]], showed in 1874 that there is more than one kind of infinity, specifically that the collection of all [[natural number]]s and the collection of all [[real number]]s, while both infinite, are not equinumerous (see [[Cantor's first uncountability proof]]). In a controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all [[rational numbers]] are equinumerous, and that the [[Cartesian product]] of even a [[countably infinite]] number of copies of the real numbers is equinumerous to a single copy of the real numbers. [[Cantor's theorem]] from 1891 implies that no set is equinumerous to its [[power set]].<ref name="suppes"/> This allows the definition of greater and greater infinite sets starting from a single infinite set.
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| Equinumerous finite sets have the same number of elements. Equinumerosity has the characteristic properties of an [[equivalence relation]].<ref name="suppes"/> Equinumerous sets are said to have the same [[cardinality]],<ref>{{cite book |title=Elements of Set Theory |last=Enderton |first=Herbert |authorlink=Herbert Enderton |publisher=Academic Press Inc. |year=1977 |isbn=0-12-238440-7}}</ref> and the [[cardinal number]] of a set is the [[equivalence class]] of all sets equinumerous to it.<ref name="suppes"/> The statement that any two sets are either equinumerous or one has a smaller cardinality than the other is equivalent to the [[axiom of choice]].<ref name="jech"/> Unlike [[finite set]]s, some infinite sets are equinumerous to [[proper subset]]s of themselves.<ref name="suppes"/>
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| == Reflexivity, symmetry, and transitivity ==
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| Equinumerosity has the characteristic properties of an [[equivalence relation]] ([[reflexive relation|reflexivity]], [[symmetric relation|symmetry]], and [[transitive relation|transitivity]]):<ref name="suppes">{{cite book |title=Axiomatic Set Theory |last=Suppes |first=Patrick |authorlink=Patrick Suppes |publisher=Dover |year=1972 |origyear=originally published by D. van Nostrand Company in 1960 |isbn=0486616304}}</ref>
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| ;Reflexivity: Given a set ''A'', the [[identity function]] on ''A'' is a bijection from ''A'' to itself, showing that every set ''A'' is equinumerous to itself: {{nowrap|''A'' ~ ''A''}}.
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| ;Symmetry: For every bijection between two sets ''A'' and ''B'' there exists an [[inverse function]] which is a bijection between ''B'' and ''A'', implying that if a set ''A'' is equinumerous to a set ''B'' then ''B'' is also equinumerous to ''A'': {{nowrap|''A'' ~ ''B''}} implies {{nowrap|''B'' ~ ''A''}}.
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| ;Transitivity: Given three sets ''A'', ''B'' and ''C'' with two bijections {{nowrap|''f'' : ''A'' → ''B''}} and {{nowrap|''g'' : ''B'' → ''C''}}, the [[composition (mathematics)|composition]] {{nowrap|''g'' ∘ ''f''}} of these bijections is a bijection from ''A'' to ''C'', so if ''A'' and ''B'' are equinumerous and ''B'' and ''C'' are equinumerous then ''A'' and ''C'' are equinumerous: {{nowrap|''A'' ~ ''B''}} and {{nowrap|''B'' ~ ''C''}} together imply {{nowrap|''A'' ~ ''C''}}.
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| Equinumerosity is not usually considered an equivalence relation, because [[binary relation|relations]] are by definition restricted to sets (a binary relation on a set ''A'' is a [[subset]] of the [[Cartesian product]] {{nowrap|''A'' × ''A''}}), and there is no [[set of all sets]] in [[Zermelo–Fraenkel set theory]], the standard form of [[axiomatic set theory]]. In some other systems of axiomatic set theory, e.g. [[Von Neumann–Bernays–Gödel set theory]] and [[Morse–Kelley set theory]], relations are extended to [[class (mathematics)|classes]].
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| == Cardinality ==
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| Equinumerous sets are said to have the same [[cardinality]]. The cardinality of a set ''X'' is a measure of the "number of elements of the set" and can be defined as the [[equivalence class]] of all sets equinumerous to ''X''.<ref name="suppes"/> This definition is problematic in [[Zermelo–Fraenkel set theory]], because the equivalence class of a [[empty set|non-empty set]] is too large to be a set. Instead one tries to assign a representative set to each equivalence class ([[cardinal assignment]]).
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| A set ''A'' is said to have cardinality smaller than or equal to the cardinality of a set ''B'' if there exists a [[one-to-one function]] (an injection) from ''A'' into ''B''. This is denoted {{nowrap begin}}|''A''| ≤ |''B''|.{{nowrap end}} If ''A'' and ''B'' are not equinumerous, then the cardinality of ''A'' is said to be strictly smaller than the cardinality of ''B''. This is denoted {{nowrap begin}}|''A''| < |''B''|.{{nowrap end}} The [[axiom of choice]] is equivalent to the statement that any two sets are either equinumerous, or one has a strictly smaller cardinality than the other.<ref name="jech">{{cite book |title=The Axiom of Choice |last=Jech |first=Thomas J. |authorlink=Thomas Jech |year=2008 |publisher=Dover |origyear=Originally published by North–Holland in 1973 |isbn=978-0-486-46624-8}}</ref> This implies the [[law of trichotomy]] for [[cardinal number]]s.<ref name="suppes"/>
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| The [[Cantor–Bernstein–Schröder theorem]] states that any two sets ''A'' and ''B'' for which there exist two one-to-one functions {{nowrap|''f'' : ''A'' → ''B''}} and {{nowrap|''g'' : ''B'' → ''A''}} are equinumerous: if {{nowrap begin}}|''A''| ≤ |''B''|{{nowrap end}} and {{nowrap begin}}|''B''| ≤ |''A''|,{{nowrap end}} then {{nowrap begin}}|''A''| = |''B''|.{{nowrap end}}<ref name="suppes"/><ref name="jech"/> This theorem does not rely on the [[axiom of choice]].
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| == Compatibility with set operations ==
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| Equinumerosity is compatible with the [[basic set operations]] in a way that allows the definition of [[cardinal arithmetic]].<ref name="suppes"/> Specifically, equinumerosity is compatible with [[disjoint union]]s: Given four sets ''A'', ''B'', ''C'' and ''D'' with ''A'' and ''C'' on the one hand and ''B'' and ''D'' on the other hand [[pairwise disjoint]] and with {{nowrap|''A'' ~ ''B''}} and {{nowrap|''C'' ~ ''D''}} then {{nowrap|''A'' ∪ ''C'' ~ ''B'' ∪ ''D''.}} This is used to justify the definition of [[cardinal addition]].
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| Furthermore, equinumerosity is compatible with [[cartesian product]]s:
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| * If {{nowrap|''A'' ~ ''B''}} and {{nowrap|''C'' ~ ''D''}} then {{nowrap|''A'' × ''C'' ~ ''B'' × ''D''.}}
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| * ''A'' × ''B'' ~ ''B'' × ''A''
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| * (''A'' × ''B'') × ''C'' ~ ''A'' × (''B'' × ''C'')
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| These properties are used to justify [[cardinal multiplication]].
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| Exponentiation:
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| * If ''A'' ~ ''B'' and ''C'' ~ ''D'' then {{nowrap|''A''<sup>''C''</sup> ~ ''B''<sup>''D''</sup>.}} (Here ''X''<sup>''Y''</sup> denotes the set of all functions from ''Y'' to ''X''.)
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| * ''A''<sup>''B'' ∪ ''C''</sup> ~ ''A''<sup>''B''</sup> × ''A''<sup>''C''</sup> for disjoint ''B'' and ''C''.
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| * (''A'' × ''B'')<sup>''C''</sup> ~ ''A''<sup>''C''</sup> × ''B''<sup>''C''</sup>
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| * (''A''<sup>''B''</sup>)<sup>''C''</sup> ~ ''A''<sup>''B''×''C''</sup>
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| These properties are used to justify [[cardinal exponentiation]].
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| Furthermore, the [[power set]] of a given set ''A'' (the set of all [[subset]]s of ''A'') is equinumerous to the set 2<sup>''A''</sup>, the set of all functions from the set ''A'' to a set containing exactly two elements.
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| == Cantor's theorem ==
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| [[Cantor's theorem]] implies that no set is equinumerous to its [[power set]] (the set of all its [[subset]]s).<ref name="suppes"/> This holds even for [[infinite set]]s. Specifically, the power set of a [[countably infinite set]] is an [[uncountable set]].
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| Assuming the existence of an infinite set '''N''' consisting of all [[natural number]]s and assuming the existence of the power set of any given set allows the definition of a sequence '''N''', ''P''('''N'''), ''P''(''P''('''N''')), {{nowrap|''P''(''P''(''P''('''N'''))), …}} of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this sequence strictly exceeds the cardinality of the set preceding it, leading to greater and greater infinite sets.
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| Cantor's work was harshly criticized by some of his contemporaries, e.g. by [[Leopold Kronecker]], who strongly adhered to a [[finitism|finitist]]<ref name="tiles">{{cite book |last=Tiles |first=Mary |authorlink=Mary Tiles |title=The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise |year=2004 |origyear=Originally published by Basil Blackwell Ltd. in 1989 |publisher=Dover |isbn=978-0486435206}}</ref> [[philosophy of mathematics]] and rejected the idea that numbers can form an actual, completed totality (an [[actual infinity]]). However, Cantor's ideas were defended by others, e.g. by [[Richard Dedekind]], and ultimately were largely accepted, strongly supported by [[David Hilbert]]. See [[Controversy over Cantor's theory]].
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| Within the framework of [[Zermelo–Fraenkel set theory]], the [[axiom of power set]] guarantees the existence of the power set of any given set. Furthermore, the [[axiom of infinity]] guarantees the existence of at least one infinite set, namely a set containing the natural numbers. There are [[alternative set theory|alternative set theories]], e.g. "[[general set theory]]" (GST), [[Kripke–Platek set theory]], and [[pocket set theory]] (PST), that deliberately omit the axiom of power set and the axiom of infinity and do not allow the definition of the infinite hierarchy of infinites proposed by Cantor.
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| The cardinalities corresponding to the sets '''N''', ''P''('''N'''), ''P''(''P''('''N''')), {{nowrap|''P''(''P''(''P''('''N'''))), …}} are the [[beth number]]s <math>\beth_0</math>, <math>\beth_1</math>, <math>\beth_2</math>, {{nowrap|<math>\beth_3</math>, …,}} with the first beth number <math>\beth_0</math> being equal to <math>\aleph_0</math> ([[aleph naught]]), the cardinality of any countably infinite set, and the second beth number <math>\beth_1</math> being equal to <math>\mathfrak c</math>, the [[cardinality of the continuum]].
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| == Dedekind-infinite sets ==
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| A given set may be equinumerous to some [[proper subset]] of itself, e.g. the set of [[natural number]]s is equinumerous to the set of even natural numbers. Such a set is called [[Dedekind-infinite]].<ref name="suppes"/><ref name="jech"/>
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| Some weak variant of the [[axiom of choice]] (AC) is needed to show that a set that is not Dedekind-infinite is [[finite set|finite]] in the sense of having a finite number of elements. The [[axiom]]s of [[Zermelo–Fraenkel set theory]] without the axiom of choice (ZF) are not strong enough to prove that every [[infinite set]] is Dedekind-infinite, but e.g. the axioms of Zermelo–Fraenkel set theory without the axiom of choice but with the [[axiom of countable choice]] ({{nowrap|ZF + AC<sub>ω</sub>}}) are strong enough.<ref name="herrlich">{{cite book |title=Axiom of Choice |last=Herrlich | first=Horst |year=2006 |publisher=Springer-Verlag |series=Lecture Notes in Mathematics 1876 |isbn=978-3540309895}}</ref> Other definitions of finiteness and infiniteness of sets do not require the axiom of choice for this.<ref name="suppes"/>
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| == Categorial definition ==
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| In [[category of sets|Set]], the [[category (category theory)|category]] of all sets with [[function (mathematics)|function]]s as morphisms, an [[isomorphism]] between two sets is precisely a bijection, and two sets are equinumerous precisely if they are isomorphic in this category.
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| == See also ==
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| * [[Combinatorial class]]
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| * [[Hume's principle]]
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| == References ==
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| {{reflist}}
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| [[Category:Basic concepts in infinite set theory]]
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| [[Category:Cardinal numbers]]
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| [[de:Mächtigkeit (Mathematik)#Gleichmächtigkeit, Mächtigkeit]]
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Jerrie Swoboda is what somebody can call me along with I totally dig that name. What me and my family genuinely like is acting but All of us can't make it all of my profession really. The job I've been taking up for years is a people manager. Guam is where I've always been living. You can sometimes find my website here: http://prometeu.net
Here is my webpage - clash of clans trainer