## You are here

HomeSchr\"oder-Bernstein theorem

## Primary tabs

# 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).

Related:

AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective, ProofOfSchroederBernsteinTheoremUsingTarskiKnasterTheorem

Synonym:

Schroeder-Bernstein theorem, Cantor-Schroeder-Bernstein theorem, Cantor-Schr\"oder-Bernstein theorem, Cantor-Bernstein theorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Sep 17

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella