CSM25 - The Cantor-Schröder-Bernstein Theorem

Given two finite sets, questions about the existence of different types of functions between these two sets are easy to solve, as there are finitely many such functions and so one many simply enumerate them. Further more, if one has two injective functions then it is intuitively obvious that both such functions are bijections (although not necessarily the inverses of each other). However, when one extends this to the infinite, the question becomes much more problematic. For instance, consider the function f : N → N, where f(x) = 2 ∗ x. This function is injective, from N to N and also trivially in the reverse direction, but f itself is not a bijection. This is a shame, as such injective functions are often much easier to find than a bijection between two sets. However, it turns out that the existence of such functions is sufficient to prove the existence of a bijection :