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).

Title	Schröder-Bernstein theorem
Canonical name	SchroderBernsteinTheorem
Date of creation	2013-03-22 12:21:46
Last modified on	2013-03-22 12:21:46
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Theorem
Classification	msc 03E10
Synonym	Schroeder-Bernstein theorem
Synonym	Cantor-Schroeder-Bernstein theorem
Synonym	Cantor-Schröder-Bernstein theorem
Synonym	Cantor-Bernstein theorem
Related topic	AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective
Related topic	ProofOfSchroederBernsteinTheoremUsingTarskiKnasterTheorem