Definition: A set
is called infinite if it has a proper subset which is bijective to Definition: is a proper subset of ( ) if is nonempty (there is some element in that is not in )
- If
has elements, then any set bijective to also must have elements
Example: Prove that
is an infinite set. Let . . This function is both injective and surjective, and is therefore bijective (each input maps perfectly to an output). is a proper subset of . For example, Thus, is infinite.
- Note that a function being Injective, surjective, bijective functions depend on the domain and the codomain. Changing them might change them might change their status.