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Schröder-Bernstein theorem (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).



"Schröder-Bernstein theorem" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: an injection between two finite sets of the same cardinality is bijective, proof of Schroeder-Bernstein theorem using Tarski-Knaster theorem

Other names:  Schroeder-Bernstein theorem, Cantor-Schroeder-Bernstein theorem, Cantor-Schröder-Bernstein theorem, Cantor-Bernstein theorem

Attachments:
proof of Schroeder-Bernstein theorem (Proof) by mps
Schröeder Bernstein Theorem: Proof (Proof) by sauravbhaumik
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Cross-references: cardinality, bijection, injections
There are 6 references to this entry.

This is version 6 of Schröder-Bernstein theorem, born on 2002-02-18, modified 2006-12-22.
Object id is 2091, canonical name is SchroederBernsteinTheorem.
Accessed 6559 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
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