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| In [[set theory]], the '''Cantor–Bernstein–Schroeder [[theorem]]''', named after [[Georg Cantor]], [[Felix Bernstein]], and [[Ernst Schröder]], states that, if there exist [[injective function]]s {{nowrap|''f'' : ''A'' → ''B''}} and {{nowrap|''g'' : ''B'' → ''A''}} between the [[Set (mathematics)|sets]] ''A'' and ''B'', then there exists a [[bijection|bijective]] function {{nowrap|''h'' : ''A'' → ''B''}}. In terms of the [[cardinality]] of the two sets, this means that if |''A''| ≤ |''B''| and |''B''| ≤ |''A''|, then |''A''| = |''B''|; that is, ''A'' and ''B'' are [[equipollent]]. This is a useful feature in the ordering of [[cardinal number]]s.
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| The theorem is also known as the Schroeder–Bernstein theorem, the Cantor–Bernstein theorem, or the Cantor–Schroeder–Bernstein theorem.
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| An important feature of this theorem is that it does not rely on the [[axiom of choice]]. However, its various proofs are [[Constructive proof|non-constructive]], as they depend on the [[law of excluded middle]], and therefore rejected by [[intuitionist]]s.<ref>{{cite book |title=Mathematics and Logic in History and in Contemporary Thought |author=Ettore Carruccio |publisher=Transaction Publishers |year=2006 |page=354 |isbn=978-0-202-30850-0}}</ref>
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| [[Image:Cantor-Bernstein.png|thumb|400px|König's definition of a bijection {{color|#00c000|''h''}}:''A''→''B'' from given example injections {{color|#c00000|''f''}}:''A''→''B'' and {{color|#0000c0|''g''}}:''B''→''A''. An element in ''A'' and ''B'' is denoted by a number and a letter, respectively. The sequence 3→e→6→… is an ''A''-stopper, leading to the defintitions {{color|#00c000|''h''}}(3)={{color|#c00000|''f''}}(3)=e, {{color|#00c000|''h''}}(6)={{color|#c00000|''f''}}(6), …. The sequence d→5→f→… is a ''B''-stopper, leading to {{color|#00c000|''h''}}(5)={{color|#0000c0|''g''}}<sup>-1</sup>(5)=d, …. The sequence …→a→1→c→4→… is doubly infinite, leading to {{color|#00c000|''h''}}(1)={{color|#c00000|''f''}}(1)=a, {{color|#00c000|''h''}}(4)={{color|#c00000|''f''}}(4)=c, …. The sequence b→2→b is cyclic, leading to {{color|#00c000|''h''}}(2)={{color|#c00000|''f''}}(2)=b.]]
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| ==Proof==
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| This proof is attributed to [[Julius König]].<ref>{{cite journal| author=J. König| title=Sur la théorie des ensembles| journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences| volume=143| pages=110–112| year=1906| url=http://gallica.bnf.fr/ark:/12148/bpt6k30977.image.f110.langEN}}</ref>
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| Assume without loss of generality that ''A'' and ''B'' are [[disjoint set|disjoint]]. For any ''a'' in ''A'' or ''b'' in ''B'' we can form a unique two-sided sequence of elements that are alternately in ''A'' and ''B'', by repeatedly applying ''<math>f</math>'' and ''<math>g</math>'' to go right and ''<math>g^{-1}</math>'' and ''<math>f^{-1}</math>'' to go left (where defined).
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| :''<math> \cdots \rightarrow f^{-1}(g^{-1}(a)) \rightarrow g^{-1}(a) \rightarrow a \rightarrow f(a) \rightarrow g(f(a)) \rightarrow \cdots </math>''
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| For any particular ''a'', this sequence may terminate to the left or not, at a point where ''<math>f^{-1}</math>'' or ''<math>g^{-1}</math>'' is not defined.
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| Call such a sequence (and all its elements) an ''A-stopper'', if it stops at an element of ''A'', or a ''B-stopper'' if it stops at an element of ''B''. Otherwise, call it ''[[doubly infinite]]'' if all the elements are distinct or ''cyclic'' if it repeats. See the picture for examples.
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| By the fact that ''<math>f</math>'' and ''<math>g</math>'' are injective functions, each ''a'' in ''A'' and ''b'' in ''B'' is in exactly one such sequence to within identity, (as if an element occurs in two sequences, all elements to the left and to the right must be the same in both, by definition). Therefore, the sequences form a [[Partition of a set|partition]] of the (disjoint) union of ''A'' and ''B''. Hence it suffices to produce a bijection between the elements of ''A'' and ''B'' in each of the sequences separately, as follows:
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| * For an ''A-stopper'', the function ''<math>f</math>'' is a bijection between its elements in ''A'' and its elements in ''B''.
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| * For a ''B-stopper'', the function ''<math>g</math>'' is a bijection between its elements in ''B'' and its elements in ''A''.
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| * For a ''doubly infinite'' sequence or a ''cyclic'' sequence, either ''<math>f</math>'' or ''<math>g</math>'' will do (<math>f</math> is used in the picture).
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| ==Another proof==
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| Below follows an alternative proof.{{citation needed|reason=Several proofs are mentioned in section 'History'. If this proof is one of them, this should be noted.|date=January 2014}}
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| '''Idea of the proof:''' Redefine ''f'' in certain points to make it surjective. At first, redefine it on the image of ''g'' for it to be the [[inverse function]] of g. However, this might destroy injectivity, so correct this problem iteratively, by making the amount of points redefined smaller, up to a minimum possible, shifting the problem "to infinity" and therefore out of sight.
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| More precisely, this means to leave ''f'' unchanged initially on ''C''<sub>0</sub> := ''A'' \ ''g''<nowiki>[</nowiki>''B''<nowiki>]</nowiki>. However, then every [[Element (mathematics)|element]] of ''f''<nowiki>[</nowiki>''C''<sub>0</sub><nowiki>]</nowiki> has two [[preimage]]s, one under ''f'' and one under ''g''<sup> –1</sup>. Therefore, leave ''f'' unchanged on the [[Union (set theory)|union]] of ''C''<sub>0</sub> and ''C''<sub>1</sub> := ''g''<nowiki>[</nowiki>''f''<nowiki>[</nowiki>''C''<sub>0</sub><nowiki>]]</nowiki>. However, then every element of ''f''<nowiki>[</nowiki>''C''<sub>1</sub><nowiki>]</nowiki> has two preimages, correct this by leaving ''f'' unchanged on the union of ''C''<sub>0</sub>, ''C''<sub>1</sub>, and ''C''<sub>2</sub> := ''g''<nowiki>[</nowiki>''f''<nowiki>[</nowiki>''C''<sub>1</sub><nowiki>]]</nowiki> and so on. Leaving ''f'' unchanged on the countable union ''C'' of ''C''<sub>0</sub> and all these ''C''<sub>''n''+1</sub> = ''g''<nowiki>[</nowiki>''f''<nowiki>[</nowiki>''C''<sub>''n''</sub><nowiki>]]</nowiki> solves the problem, because ''g''<nowiki>[</nowiki>''f''<nowiki>[</nowiki>''C''<nowiki>]]</nowiki> is a subset of ''C'' and no additional union is necessary.
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| In the alternate proof, ''C''<sub>''n''</sub> can be interpreted as the set of n-th elements of ''A-stoppers'' (starting from 0).
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| Indeed, ''C''<sub>0</sub> is the set of elements for which ''g''<sup>−1</sup> is not defined, which is the set of starting elements of ''A-stoppers'', ''C''<sub>1</sub> is the set of elements for which ''<math>f^{-1}\circ g^{-1}</math>'' is defined but ''<math>g^{-1}\circ f^{-1}\circ g^{-1}</math>'' is not, i.e. the set of second elements of ''A-stoppers'', and so on.
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| The bijection ''h'' is defined as ''f'' on ''C'' and ''g''<sup>−1</sup> everywhere else, which means ''f'' on ''A-stoppers'' and ''g''<sup>−1</sup> everywhere else, consistently with the proof below.
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| '''Proof:''' Define
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| :<math>C_0=A\setminus g[B],\qquad C_{n+1}=g[f[C_n]]\quad \mbox{ for all }n\ge 0,</math>
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| and
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| :<math>C=\bigcup_{n=0}^\infty C_n.</math>
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| Then, for every ''a'' ∈ ''A'' define
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| :<math>
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| h(a)=
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| \begin{cases}
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| f(a) & \mbox{if }a\in C, \\
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| g^{-1}(a) & \mbox{if }a\notin C.
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| \end{cases}
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| </math>
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| If ''a'' is not in ''C'', then, in particular, ''a'' is not in ''C''<sub>0</sub>. Hence ''a'' ∈ ''g''<nowiki>[</nowiki>''B''<nowiki>]</nowiki> by the definition of ''C''<sub>0</sub>. Since ''g'' is injective, its preimage ''g''<sup> –1</sup>(''a'') is therefore well defined.
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| It remains to check the following properties of the map ''h'' : ''A'' → ''B'' to verify that it is the desired bijection:
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| * '''Surjectivity:''' Consider any ''b'' ∈ ''B''. If ''b'' ∈ ''f''<nowiki>[</nowiki>''C''<nowiki>]</nowiki>, then there is an ''a'' ∈ ''C'' with ''b'' = ''f''(''a''). Hence ''b'' = ''h''(''a'') by the definition of ''h''. If ''b'' is not in ''f''<nowiki>[</nowiki>''C''<nowiki>]</nowiki>, define ''a'' = ''g''(''b''). By definition of ''C''<sub>0</sub>, this ''a'' cannot be in ''C''<sub>0</sub>. Since ''f''<nowiki>[</nowiki>''C''<sub>''n''</sub><nowiki>]</nowiki> is a subset of ''f''<nowiki>[</nowiki>''C''<nowiki>]</nowiki>, it follows that ''b'' is not in any ''f''<nowiki>[</nowiki>''C''<sub>''n''</sub><nowiki>]</nowiki>, hence ''a'' = ''g''(''b'') is not in any ''C''<sub>''n''+1</sub> = ''g''<nowiki>[</nowiki>''f''<nowiki>[</nowiki>''C''<sub>''n''</sub><nowiki>]]</nowiki> by the recursive definition of these sets. Therefore, ''a'' is not in ''C''. Then ''b'' = ''g''<sup> –1</sup>(''a'') = ''h''(''a'') by the definition of ''h''.
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| * '''Injectivity:''' Since ''f'' is injective on ''A'', which comprises ''C'', and ''g''<sup> –1</sup> is injective on ''g''<nowiki>[</nowiki>''B''<nowiki>]</nowiki>, which comprises the complement of ''C'', it suffices to show that the assumption ''f''(''c'') = ''g''<sup> –1</sup>(''a'') for ''c'' ∈ ''C'' and ''a'' ∈ ''A'' \ ''C'' leads to a contradiction (this means the original problem, the lack of injectivity mentioned in the idea of the proof above, is solved by the clever definition of ''h''). Since ''c'' ∈ ''C'', there exists an integer ''n'' ≥ 0 such that ''c'' ∈ ''C''<sub>''n''</sub>. Hence ''g''(''f''(''c'')) is in ''C''<sub>''n''+1</sub> and therefore in ''C'', too. However, ''g''(''f''(''c'')) = ''g''(''g''<sup> –1</sup>(''a'')) = ''a'' is not in ''C'' — contradiction.
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| Note that the above definition of ''h'' is nonconstructive, in the sense that there exists no ''general'' method to decide in a finite number of steps, for any given sets ''A'' and ''B'' and injections ''f'' and ''g'', whether an element ''a'' of ''A'' does not lie in ''C''. For special sets and maps this might, of course, be possible.
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| ==Original proof==
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| An earlier proof by [[Georg Cantor|Cantor]] relied, in effect, on the [[axiom of choice]] by inferring the result as a [[corollary]] of the [[well-ordering theorem]].<ref>{{cite journal |author=Georg Cantor |title=Beiträge zur Begründung der transfiniten Mengenlehre (1) |url=http://gdz.sub.uni-goettingen.de/index.php?id=img&no_cache=1&IDDOC=36218&IDDOC=36218&branch=&L=1|journal=Mathematische Annalen |volume=46 | |page=481-512 |year=1895}}<BR>
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| {{cite journal |author=Georg Cantor |title=Beiträge zur Begründung der transfiniten Mengenlehre (2) |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0049&DMDID=DMDLOG_0024&L=1 |journal=Mathematische Annalen |volume=49 |page=207-246 |year= 1897}}</ref> The argument given above{{clarify|reason=Probably, König's proof is meant?|date=January 2014}} shows that the result can be proved without using the axiom of choice.
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| Furthermore, there is a simple proof which uses [[Knaster–Tarski theorem|Tarski's fixed point theorem]].<ref>R. Uhl, "[http://mathworld.wolfram.com/TarskisFixedPointTheorem.html Tarski's Fixed Point Theorem]", from ''MathWorld''–a Wolfram Web Resource, created by Eric W. Weisstein. (Example 3)</ref>
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| == History ==
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| As it is often the case in mathematics, the name of this theorem does not truly reflect its history.
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| The traditional name "Schröder-Bernstein" is based on two proofs published independently in 1898.
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| Cantor is often added because he first stated the theorem in 1895,
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| while Schröder's name is often omitted because his proof turned out to be flawed
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| while the name of the mathematician who first proved it is not connected with the theorem.
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| [[File:Cantor1895.MA46.484-BC.jpg|thumb|right|Georg Cantor (1895) states the theorem (B.)]]
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| In reality, the history was more complicated:
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| * '''1887''' [[Richard Dedekind]] proves the theorem but does not publish it.
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| * '''1895''' [[Georg Cantor]] states the theorem in his first paper on set theory and transfinite numbers (as an easy consequence of the linear order of cardinal numbers which he was going to prove later).
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| * '''1896''' [[Ernst Schröder]] announces a proof (as a corollary of a more general statement).
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| * '''1897''' [[Felix Bernstein]], a young student in Cantor's Seminar, presents his proof.
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| * '''1897''' After a visit by Bernstein, Dedekind independently proves it a second time.
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| * '''1898''' Bernstein's proof is published by [[Émile Borel]] in his book on functions. (Communicated by Cantor at the 1897 congress in Zürich.)
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| Both proofs of Dedekind are based on his famous memoir ''Was sind und was sollen die Zahlen?''
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| and derive it as a corollary of a proposition equivalent to statement C in Cantor's paper:
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| :<math>
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| A \subset B \subset C \quad\textrm{and}\quad |A|=|C| \qquad\Rightarrow\qquad |A|=|B|=|C|
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| </math>
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| Cantor observed this property as early as 1882/83 during his studies in set theory and transfinite numbers
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| and therefore (implicitly) relying on the [[Axiom of Choice]].
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| == See also ==
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| * [[Myhill isomorphism theorem]]
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| * [[Schröder–Bernstein theorem for measurable spaces]]
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| * [[Schröder–Bernstein theorems for operator algebras]]
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| * [[Schröder–Bernstein property]]
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| == Notes ==
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| <references/>
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| == References ==
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| {{Citizendium|title=Schröder-Bernstein_theorem}} Peter Schmitt contributed the History section to Citizendium which TakuyaMurata copied into this article.
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| * ''Proofs from THE BOOK'', p. 90. ISBN 3-540-40460-0
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| * {{planetmath reference|id=3156|title=Proof of the Bernstein–Schroeder theorem}}
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| *[http://www.mathpath.org/proof/Sch-Bern/proofofS-B.htm MathPath – Explanation of and remarks on the proof of Cantor–Bernstein Theorem ]
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| * [http://www.hinkis.org/HTML_pages/CBT_papers.html Papers on the history of the Cantor–Bernstein theorem ]
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| *{{nlab|id=Cantor-Schroeder-Bernstein+theorem|title=Cantor-Schroeder-Bernstein theorem}}
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| ==External links==
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| *Citizendium: [http://en.citizendium.org/wiki/Schr%C3%B6der-Bernstein_theorem Schröder-Bernstein theorem] and [http://en.citizendium.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_property Schröder–Bernstein property]
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| *{{MathWorld|title=Schröder-Bernstein Theorem|urlname=Schroeder-BernsteinTheorem}}
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| *{{PlanetMath|urlname=schroderbernsteintheorem|title=Schröder-Bernstein theorem}}
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| {{DEFAULTSORT:Cantor-Bernstein-Schroeder Theorem}}
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| [[Category:Theorems in the foundations of mathematics]]
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| [[Category:Cardinal numbers]]
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| [[Category:Articles containing proofs]]
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