Schr\"oder-Bernstein theorem SchroederBernsteinTheorem 2002-02-18 01:25:51 2006-12-22 07:13:20 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{theorem} \PMlinkescapeword{between} 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 Schr\"oder-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).