# Schröder–Bernstein theorem

In set theory, the **Schröder–Bernstein theorem** states that, if there exist injective functions *f* : *A* → *B* and *g* : *B* → *A* between the sets *A* and *B*, then there exists a bijective function *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 equipotent. This is a useful feature in the ordering of cardinal numbers.

This theorem does not rely on the axiom of choice. However, its various proofs are non-constructive, as they depend on the law of excluded middle, and are therefore rejected by intuitionists.^{[1]}

The theorem is named after Felix Bernstein and Ernst Schröder. It is also known as **Cantor–Bernstein theorem**, or **Cantor–Schröder–Bernstein**, after Georg Cantor who first published it without proof.

## Contents

## Proof[edit]

The following proof is attributed to Julius König.^{[2]}

Assume without loss of generality that *A* and *B* are 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 and to go from *A* to *B* and and to go from *B* to *A* (where defined).

For any particular *a*, this sequence may terminate to the left or not, at a point where or is not defined.

By the fact that and are injective functions, each *a* in *A* and *b* in *B* is in exactly one such sequence to within identity: if an element occurs in two sequences, all elements to the left and to the right must be the same in both, by the definition of the sequences. Therefore, the sequences form a 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:

Call a sequence 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.

- For an
*A-stopper*, the function*is a bijection between its elements in**A*and its elements in*B*. - For a
*B-stopper*, the function*is a bijection between its elements in**B*and its elements in*A*. - For a
*doubly infinite*sequence or a*cyclic*sequence, either*or**will do ( is used in the picture).*

## Original proof[edit]

An earlier proof by Cantor relied, in effect, on the axiom of choice by inferring the result as a corollary of the well-ordering theorem.^{[3]} The argument given above shows that the result can be proved without using the axiom of choice. Note however that the principle of excluded middle is used to do the analysis into cases, so this proof does not work in non-classical logic.

There is also a proof which uses Tarski's fixed point theorem.^{[4]}

## History[edit]

The traditional name "Schröder–Bernstein" is based on two proofs published independently in 1898.
Cantor is often added because he first stated the theorem in 1887,
while Schröder's name is often omitted because his proof turned out to be flawed
while the name of Richard Dedekind, who first proved it, is not connected with the theorem.
According to Bernstein, Cantor had suggested the name *equivalence theorem* (Äquivalenzsatz).^{[5]}

**1887****Cantor**publishes the theorem, however without proof.^{[6]}^{[5]}**1887**On July 11,**Dedekind**proves the theorem (not relying on the axiom of choice)^{[7]}but neither publishes his proof nor tells Cantor about it. Ernst Zermelo discovered Dedekind's proof and in 1908^{[8]}he publishes his own proof based on the*chain theory*from Dedekind's paper*Was sind und was sollen die Zahlen?*^{[5]}^{[9]}**1895****Cantor**states the theorem in his first paper on set theory and transfinite numbers. He obtains it as an easy consequence of the linear order of cardinal numbers.^{[10]}^{[11]}However, he couldn't prove the latter theorem, which is shown in 1915 to be equivalent to the axiom of choice by Friedrich Moritz Hartogs.^{[5]}^{[12]}**1896****Schröder**announces a proof (as a corollary of a theorem by Jevons).^{[13]}**1896****Schröder**publishes a proof sketch^{[14]}which, however, is shown to be faulty by Alwin Reinhold Korselt in 1911^{[15]}(confirmed by Schröder).^{[5]}^{[16]}**1897****Bernstein**, a 19 years old student in Cantor's Seminar, presents his proof.^{[17]}^{[18]}**1897**Almost simultaneously, but independently,**Schröder**finds a proof.^{[17]}^{[18]}**1897**After a visit by Bernstein,**Dedekind**independently proves the theorem a second time.**1898****Bernstein'**s proof (not relying on the axiom of choice) is published by Émile Borel in his book on functions.^{[19]}(Communicated by Cantor at the 1897 International Congress of Mathematicians in Zürich.) In the same year, the proof also appears in**Bernstein'**s dissertation.^{[20]}^{[5]}

Both proofs of Dedekind are based on his famous memoir *Was sind und was sollen die Zahlen?* and derive it as a corollary of a proposition equivalent to statement C in Cantor's paper,^{[10]} which reads *A* ⊆ *B* ⊆ *C* and |*A*| = |*C*| implies |*A*| = |*B*| = |*C*|. Cantor observed this property as early as 1882/83 during his studies in set theory and transfinite numbers and was therefore (implicitly) relying on the Axiom of Choice.

## Statement C and a transparent proof[edit]

Statement C is the special case of the Schröder–Bernstein theorem where the second function *g* is the identity (and, hence, the second set *B* is a subset of *A*).

It is easy to see that statement C implies the general form of the theorem (with *g* and *B* arbitrary):

Assume that *ƒ* injects *A* into *B* and *g* injects *B* into *A*. Then their composition *gf* injects *A* into *g*[*B*]. However, *g*[*B*] is a subset of *A*. Thus, from statement *C*, we obtain that *A* and *g*[*B*] are equipotent. Obviously, *B* and *g*[*B*] are equipotent as well. It follows, that *A* and *B* are equipotent.

A proof of statement *C* can be obtained by translating the König proof for the general case above to the present situation. The result becomes much more transparent than the original, and looks as follows.

Assume that *ƒ* injects *A* into its subset *C*. Consider the subset *D* of *A* (the set of *A*-stoppers, in the König terminology above) which is the union of the infinitely many sets *A* − *C*, *ƒ*[*A* − *C*], *ƒƒ*[*A* − *C*]], ...

Consider the function from *A* to *C* that (i) maps elements *a* of *D* to their *ƒ*-image *ƒ*(*a*), whereas (ii) on *A* − *D* it acts as the identity, mapping elements to themselves.

We claim that this function is a bijection from *A* onto *C*. For this, it should be verified that every element of *C* has exactly one original. However, if it belongs to *D*, it has one original which also is an *f*-original, and if it doesn't belong to *D* the only original is the element itself.

## See also[edit]

- Myhill isomorphism theorem
- Schröder–Bernstein theorem for measurable spaces
- Schröder–Bernstein theorems for operator algebras
- Schröder–Bernstein property

## Notes[edit]

**^**Ettore Carruccio (2006).*Mathematics and Logic in History and in Contemporary Thought*. Transaction Publishers. p. 354. ISBN 978-0-202-30850-0.**^**J. König (1906). "Sur la théorie des ensembles".*Comptes rendus hebdomadaires des séances de l'Académie des sciences*.**143**: 110–112.**^**Georg Cantor (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)".*Mathematische Annalen*.**46**: 481–512. doi:10.1007/bf02124929.

Georg Cantor (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)".*Mathematische Annalen*.**49**: 207–246. doi:10.1007/bf01444205.**^**R. Uhl, "Tarski's Fixed Point Theorem", from*MathWorld*–a Wolfram Web Resource, created by Eric W. Weisstein. (Example 3)- ^
^{a}^{b}^{c}^{d}^{e}^{f}Felix Hausdorff (2002), Egbert Brieskorn; Srishti D. Chatterji; et al., eds.,*Grundzüge der Mengenlehre*(1. ed.), Berlin/Heidelberg: Springer, p. 587, ISBN 3-540-42224-2 – Original edition (1914) - ^
^{a}^{b}Georg Cantor (1887), "Mitteilungen zur Lehre vom Transfiniten",*Zeitschrift für Philosophie und philosophische Kritik*,**91**: 81–125

Reprinted in: Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo, eds.,*Gesammelte Abhandlungen mathematischen und philosophischen Inhalts*, Berlin: Springer, pp. 378–439 Here: p.413 bottom **^**Richard Dedekind (1932), Robert Fricke; Emmy Noether; Øystein Ore, eds.,*Gesammelte mathematische Werke*,**3**, Braunschweig: Friedr. Vieweg & Sohn, pp. 447–449 (Ch.62)**^**Ernst Zermelo (1908), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal, eds., "Untersuchungen über die Grundlagen der Mengenlehre I",*Mathematische Annalen*, Leipzig: B. G. Teubner,**65**(2): 261–281, here: p.271–272, doi:10.1007/bf01449999, ISSN 0025-5831**^**Richard Dedekind (1888),*Was sind und was sollen die Zahlen?*(2., unchanged (1893) ed.), Braunschweig: Friedr. Vieweg & Sohn- ^
^{a}^{b}Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo, eds.,*Gesammelte Abhandlungen mathematischen und philosophischen Inhalts*, Berlin: Springer, pp. 285 ("Satz B") **^**Georg Cantor (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)".*Mathematische Annalen*.**46**: 481–512 (Theorem see "Satz B", p.484). doi:10.1007/bf02124929.

(Georg Cantor (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)".*Mathematische Annalen*.**49**: 207–246. doi:10.1007/bf01444205.)**^**Friedrich M. Hartogs (1915), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal, eds., "Über das Problem der Wohlordnung",*Mathematische Annalen*, Leipzig: B. G. Teubner,**76**(4): 438–443, doi:10.1007/bf01458215, ISSN 0025-5831**^**Ernst Schröder (1896). "Über G. Cantorsche Sätze".*Jahresbericht der Deutschen Mathematiker-Vereinigung*.**5**: 81–82.**^**Ernst Schröder (1898), Kaiserliche Leopoldino-Carolinische Deutsche Akademie der Naturforscher, ed., "Ueber zwei Definitionen der Endlichkeit und G. Cantor'sche Sätze",*Nova Acta*, Halle a. S.: Johann Ambrosius Barth Verlag,**71**(6): 303–376 (proof: p.336–344)**^**Alwin R. Korselt (1911), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal, eds., "Über einen Beweis des Äquivalenzsatzes",*Mathematische Annalen*, Leipzig: B. G. Teubner,**70**(2): 294–296, doi:10.1007/bf01461161, ISSN 0025-5831**^**Korselt (1911), p.295- ^
^{a}^{b}Oliver Deiser (2010),*Einführung in die Mengenlehre – Die Mengenlehre Georg Cantors und ihre Axiomatisierung durch Ernst Zermelo*(3rd, corrected ed.), Berlin/Heidelberg: Springer, pp. 71, 501, doi:10.1007/978-3-642-01445-1, ISBN 978-3-642-01444-4 - ^
^{a}^{b}Patrick Suppes (1972),*Axiomatic Set Theory*(1. ed.), New York: Dover Publications, pp. 95 , f., ISBN 0-486-61630-4 **^**Émile Borel (1898),*Leçons sur la théorie des fonctions*, Paris: Gauthier-Villars et fils, pp. 103 , ff.**^**Felix Bernstein (1901),*Untersuchungen aus der Mengenlehre*, Halle a. S.: Buchdruckerei des Waisenhauses

Reprinted in: Felix Bernstein (1905), Felix Klein; Walther von Dyck; David Hilbert, eds., "Untersuchungen aus der Mengenlehre",*Mathematische Annalen*, Leipzig: B. G. Teubner,**61**(1): 117–155, (Theorem see "Satz 1" on p.121), doi:10.1007/bf01457734, ISSN 0025-5831

## References[edit]

- Martin Aigner & Gunter M. Ziegler (1998)
*Proofs from THE BOOK*, § 3 Analysis: Sets and functions, Springer books MR1723092, fifth edition 2014 MR3288091, sixth edition 2018 MR3823190 - Hinkis, Arie (2013),
*Proofs of the Cantor-Bernstein theorem. A mathematical excursion*, Science Networks. Historical Studies,**45**, Heidelberg: Birkhäuser/Springer, doi:10.1007/978-3-0348-0224-6, ISBN 978-3-0348-0223-9, MR 3026479 - Míchaél Ó Searcóid (2013) "On the history and mathematics of the equivalence theorem", Mathematical Proceedings of the Royal Irish Academy 113A: 151–68 doi:10.3311/PRIA.2013.113.14 Jstor link

## External links[edit]

- Weisstein, Eric W. "Schröder-Bernstein Theorem".
*MathWorld*. - Cantor-Schroeder-Bernstein theorem in
*nLab* - Cantor-Bernstein’s Theorem in a Semiring by Marcel Crabbé.
*This article incorporates material from the Citizendium article "Schröder-Bernstein_theorem", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.*