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.
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).
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. 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.
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).
- 1887 Cantor publishes the theorem, however without proof.
- 1887 On July 11, Dedekind proves the theorem (not relying on the axiom of choice) but neither publishes his proof nor tells Cantor about it. Ernst Zermelo discovered Dedekind's proof and in 1908 he publishes his own proof based on the chain theory from Dedekind's paper Was sind und was sollen die Zahlen?
- 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. 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.
- 1896 Schröder announces a proof (as a corollary of a theorem by Jevons).
- 1896 Schröder publishes a proof sketch which, however, is shown to be faulty by Alwin Reinhold Korselt in 1911 (confirmed by Schröder).
- 1897 Bernstein, a 19 years old student in Cantor's Seminar, presents his proof.
- 1897 Almost simultaneously, but independently, Schröder finds a proof.
- 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. (Communicated by Cantor at the 1897 International Congress of Mathematicians in Zürich.) In the same year, the proof also appears in Bernstein's dissertation.
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, 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
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.
- Myhill isomorphism theorem
- Schröder–Bernstein theorem for measurable spaces
- Schröder–Bernstein theorems for operator algebras
- Schröder–Bernstein property
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Georg Cantor (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49: 207–246. doi:10.1007/bf01444205.
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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
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(Georg Cantor (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49: 207–246. doi:10.1007/bf01444205.)
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- 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.