Prove that if a set is countably or uncountably infinite, then there is a proper subset and a bijection .
Consider the case in which the set is countably infinite. Then . Then there is a set such that . And so there is an injective function . Then is an injection and also a surjection which implies that it is a bijection.
Consider case in which is uncountably infinite. Then . Suppose that is a set such that .
Now how do I proceed?
A set is finite if it is equiponent to a natural number. A set is infinite if it is not finite (no seriously, that is how we define it). If is finite then there is no proper subset with . Thus, if has a proper subset equiponent to it then it means is infinite. We may wonder if is a set such that it is has a proper subset equiponent to it, then is it infinite (this property is Dedekind infinite)? Phrased more differently are infinite sets equivalent to Dedekind infinite sets? With the Axiom of Choice this is true. Let be an infinite set. And let . Then since it means . Thus, is equipotent to .