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# Schröder-Bernstein theorem

Let $S$ and $T$ be sets. If there are injections $S\to T$ and $T\to S$, then there is a bijection $S\to T$.

The Schröder-Bernstein theorem is useful for proving many results about cardinality, since it replaces one hard problem (finding a bijection between $S$ and $T$) with two generally easier problems (finding two injections).

Related:

AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective, ProofOfSchroederBernsteinTheoremUsingTarskiKnasterTheorem

Synonym:

Schroeder-Bernstein theorem, Cantor-Schroeder-Bernstein theorem, Cantor-Schr\"oder-Bernstein theorem, Cantor-Bernstein theorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E10*no label found*

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new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith