◀ ▲ ▶Branches / Set-theory / Definition: Comparison of Cardinal Numbers

The power of Cantor's idea of a cardinal number lies in the fact that it is good for classifying and comparing not only finite but also infinite sets with respect to the number of their elements.

Definition: Comparison of Cardinal Numbers

Let \(A\) and \(B\) be sets. We define for the cardinals \(|A|\) and \(|B|\):

• $|A|\le |B|$, if and only if there is a injective function \(f:A\to B\).
• $|A| < |B|$, if and only if there is an injective, but no surjective function \(f:A\to B\).

Mentioned in:

Chapters: 1
Definitions: 2
Explanations: 3
Motivations: 4
Proofs: 5 6 7 8 9
Propositions: 10

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References

Bibliography

1. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition