◀ ▲ ▶Branches / Set-theory / Theorem: Distinction Between Finite and Infinite Sets Using Subsets

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 $f:X\to S.$

Table of Contents

Proofs: 1

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References

Bibliography

1. Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition