The previous concepts allow making a distinction between finite and infinite sets. In some literature, you will find this result not as a *theorem*, but as a *definition* of finite and infinite sets.

Let $X$ be a non-empty set and $S\subset X$ its proper subset. Then

- $X$ is finite, if and only if there is no injective function $f:X\to S.$
- $X$ is infinite, if and only if there is at least one injective function $f:X\to S.$

| | | | | created: 2019-09-07 15:20:26 | modified: 2019-09-07 16:18:44 | by: *bookofproofs* | references: [8297]

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[8297] **Flachsmeyer, Jürgen**: “Kombinatorik”, VEB Deutscher Verlag der Wissenschaften, 1972, Dritte Auflage