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Definition: Countable Set, Uncountable Set

A set \(D\) is called

countable, if the comparison of the cardinal numbers of $D$ and the set of natural numbers $\mathbb N$ gives us the relation $|D|\le |\mathbb N|.$ i.e. if there is an injective function $f:D\to\mathbb N.$¹.
else uncountable, i.e. if $|D|>|\mathbb N|.$

Branches: 1
Corollaries: 2
Definitions: 3 4 5 6 7 8 9 10 11
Explanations: 12 13
Parts: 14
Proofs: 15 16 17 18 19
Propositions: 20 21 22

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References

Bibliography

Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001

Footnotes

This is equivalent with saying that there is a surjective function function \(f:\mathbb N\mapsto D.\) Some books define countability by requiring a bijective function between $D$ and $\mathbb N,$ but the above definition has the advantage that it is also applicable for a finite set $D.$ Thus, all finite sets are countable. ↩