Definition – (In)finite set. – Math-reference.net

Definition

Let $S$ be a set. We say that $S$ is an infinite set if there exists a subset $S'$ of $S$ such that there exists a bijective function $f: S' \rightarrow S$ . If the set $S$ is not infinite, we say that it is finite.

Example

The function $f: S' \rightarrow S$ , given by $f(s') = 2s'$ , is a bijection between the natural numbers $\mathbb{N}$ and the even numbers. It follows that there is a subset $S'$ of $\mathbb{N}$ for which a bijection exists, and therefore $\mathbb{N}$ is an infinite set.