**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 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 $|D|>|\mathbb N|.$

^{1} 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.

| | | | | created: 2014-07-11 21:08:10 | modified: 2019-01-26 11:11:14 | by: *bookofproofs* | references: [577]

## 1.**Proposition**: Union of Countable Many Countable Sets

## 2.**Proposition**: Real Numbers are Uncountable

## 3.**Proposition**: Uncountable and Countable Subsets of Natural Numbers

(none)

[577] **Knauer Ulrich**: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

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