Schröder–Bernstein theorem

In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions $f : A \to B$ and $g : B \to A$ between the sets $A$ and $B$ , then there exists a bijective function $h : A \to B$ .

In terms of the cardinality of the two sets, this classically implies that if $| A | \leq | B |$ and $| B | \leq | A |$ , then $| A | = | B |$ ; that is, $A$ and $B$ are equipotent. This is a useful feature in the ordering of cardinal numbers.

The theorem is named after Felix Bernstein and Ernst Schröder. It is also known as the Cantor–Bernstein theorem or Cantor–Schröder–Bernstein theorem, after Georg Cantor, who first published it (albeit without proof).