# 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 |