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.

Theorem: Distinction Between Finite and Infinite Sets Using Subsets 

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]

1.Proof: (related to "Distinction Between Finite and Infinite Sets Using Subsets")