Schroeder-Bernstein Theorem

• categories: Real Analysis, Set Theory, Theorem

Let $A$ and $B$ be two sets. If there exist injective (one-to-one) functions $f:A\to B$ and $g:B\to A$, then there exists a bijective function $h:A\to B$. In other words, if $A$ can be injected into $B$ and $B$ can be injected into $A$, then $A$ and $B$ have the same cardinality (i.e., there is a one-to-one correspondence between them)

Proof Outline:
The proof involves constructing the bijection $h:A\to B$ from the injective functions $f$ and $g$. The construction typically involves partitioning the sets based on the images of $f$ and $g$, and then carefully defining $h$ on these partitions.