# 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