Math 2.0

Mathematical logic and foundations Ordinals and cardinals Theorem

Schröder–Bernstein theorem

Builds on Injective functionBijective function

Statement

Let AA and BB be sets. If there are injective functions f ⁣:A→Bf \colon A \to B and g ⁣:B→Ag \colon B \to A, then there is a bijective function h ⁣:A→Bh \colon A \to B.

Remarks

In terms of cardinals: ∣A∣≤∣B∣|A| \le |B| and ∣B∣≤∣A∣|B| \le |A| imply ∣A∣=∣B∣|A| = |B|. So the order of cardinals is antisymmetric. The proof does not use the axiom of choice. Example: [0,1][0, 1] and [0,1)[0, 1) are equinumerous, because each one injects into the other.

History

Cantor stated the theorem in 1887 without a proof. Dedekind proved it in 1887, but he did not publish the proof. Bernstein and Schröder published proofs in 1898, and the proof of Schröder had a flaw.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionSchröder–Bernstein theorem, statement, no evidence levelSchröder–Bernstein th…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Function definition
  14. Codomain definition
  15. Injective function definition
  16. Surjective function definition
  17. Bijective function definition
  18. Schröder–Bernstein theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Schröder–Bernstein_theorem, lead section
About this page
Tag0A9V6RKQG6
URIm2://0A9V6RKQG6
Aliasescantor-bernstein-theorem, cantor-schroeder-bernstein-theorem
Kindstatement, theorem
WritingW1 stated
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0A9V6RKQG6.
UsesInjective function, Bijective function
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