# Today I Learned

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

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

Given two sets $A, B$, there exists an injection $F: A \rightarrow B$ if and only if there exists a surjection $G: B \rightarrow A$. 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 $A,B$ if and only if there exist mutually bijective functions $F: A \rightarrow B, G: B \rightarrow A$ — has a couple obvious corrolaries:

• There exists a bijection between two sets $A,B$ if and only if there exist mutually surjective functions $F: A \rightarrow B, G: B \rightarrow A$. (In terms of cardinality: $|A| = |B|$ iff $|A| \geq |B|$ and $|B| \geq |A|$.)
• There exists a bijection between two sets $A,B$ if and only if there exists an injective function $F: A \rightarrow B$ as well as a surjective function $G: A \rightarrow B$. (In terms of cardinality, $|A| = |B|$ iff $|A| \geq |B|$ and $|A| \leq |B|$.)