Cardinality

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The cardinality of a set is the number of elements of that set, represented by a cardinal number. Two sets have the same cardinality if and only if there is a bijection (one-to-one and onto correspondence) between the two sets. (While this is rather trivial for finite sets, the concept is essential for infinite sets.) The cardinality of a set  

A

is denoted in one of the following ways:

$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(A)=|A|=\mathrm{\#}A.$

Examples

$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\{1,2,3,4,5,6\})=|\{1,2,3,4,5,6\}|=\mathrm{\#}\{1,2,3,4,5,6\}=6.$

$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\})=|\{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}|=\mathrm{\#}\{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}=8.$

Cardinality of infinite sets

The cardinality of the set of natural numbers defines  

ℵ0

(pronounced aleph null or aleph nought, aleph being the first letter of the Hebrew alphabet), i.e. the cardinality of countably infinite sets (enumerable sets, denumerable sets), i.e. sets for which a one-to-one and onto correspondence with the set of natural numbers can be made

${\mathrm{\aleph }}_0:=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\mathbb{\mathbb{N}})=|\mathbb{\mathbb{N}}|=\mathrm{\#}\mathbb{\mathbb{N}}.$

A set which has an injection into the natural numbers (i.e., either has a bijection to the natural numbers or is finite) is called countable. Georg Cantor proved that the the set of rational numbers are countable:

$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\mathbb{\mathbb{Q}})=|\mathbb{\mathbb{Q}}|=\mathrm{\#}\mathbb{\mathbb{Q}}={\mathrm{\aleph }}_0,$

and it can be proved with a generalization of Cantor's idea that the set of algebraic numbers is also countable:

$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\mathbb{𝔸})=|\mathbb{𝔸}|=\mathrm{\#}\mathbb{𝔸}={\mathrm{\aleph }}_0.$

The cardinality of the real numbers, called the cardinality of the continuum and denoted $\mathfrak{𝔠}$ or ${\beth }_1$, is the cardinality of the power set of the natural numbers

$\mathfrak{𝔠}:=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(\mathbb{ℝ})=|\mathbb{ℝ}|=\mathrm{\#}\mathbb{ℝ}=2^{{\mathrm{\aleph }}_0}.$

The continuum hypothesis states that there are no cardinalities between  

ℵ0

and  

2 ℵ0

, so that  

ℵ1 = 2 ℵ0

.