DEFINITION 1: A set S is finite with cardinality n∈N if there is a bijection from the set {0,1,...,n−1} to S. A set is infinite if its not finite.
THEOREM 1: The set N of natural numbers is an infinite set.
Proof: Consider the injection f:N→N defined as f(x)=3x. The range of f is a subset of the domain of f.
I understand that f(x)=3x is not surjective and thus not bijective since for example the range does not contain number 2. But what would happen if we were to define f:N→N as f(x)=x? It is a bijective function. Doesn't that make the set of natural numbers finite according to the definition? What am I missing can somebody please tell me?
Answer
No. The definition of finite is f:{0,1,...,n−1}→S is bijective.
We know f:N→N via f(n)=n is bijective, but this maps N onto N. It does not map {0,1,...,n−1} onto N.
Basically, this prove N is finite if N is finite.
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