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