# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 2: Corrolaries to the Cantor-Schröder-Bernstein Theorem

October 12, 2016

Posted by on Given two sets , there exists an injection if and only if there exists a surjection . This can easily be proved in a constructive manner.

Given this, the Cantor-Schröder-Bernstein Theorem — which states that there exists a bijection between two sets if and only if there exist mutually bijective functions — has a couple obvious corrolaries:

- There exists a bijection between two sets if and only if there exist mutually surjective functions . (In terms of cardinality: iff and .)
- There exists a bijection between two sets if and only if there exists an injective function as well as a surjective function . (In terms of cardinality, iff and .)

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