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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

is denoted in one of the following ways:

- ${\rm {card}}(A)=|A|=\#A.$

## Examples

- ${\rm {card}}(\{1,2,3,4,5,6\})=|\{1,2,3,4,5,6\}|=\#\{1,2,3,4,5,6\}=6.$

- ${\rm {card}}(\{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\})=|\{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}|=\#\{\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}=8.$

## Cardinality of infinite sets

The cardinality of the set of

natural numbers defines

(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

- $\aleph _{0}:={\rm {card}}(\mathbb {N} )=|\mathbb {N} |=\#\mathbb {N} .$

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:

- ${\rm {card}}(\mathbb {Q} )=|\mathbb {Q} |=\#\mathbb {Q} =\aleph _{0},$

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

- ${\rm {card}}({\mathbb {A} })=|{\mathbb {A} }|=\#{\mathbb {A} }=\aleph _{0}.$

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

- ${\mathfrak {c}}:={\rm {card}}(\mathbb {R} )=|\mathbb {R} |=\#\mathbb {R} =2^{\aleph _{0}}.$

The

continuum hypothesis states that there are no cardinalities between

and

, so that

.