User Contributed Dictionary
 Of a set, the number of elements it contains.
 The empty set has a cardinality of zero.
Synonyms
 (in set theory): power
Related terms
Translations
in set theory
 Czech: mohutnost , kardinalita
 Dutch: kardinalitet
 German: Mächtigkeit , Kardinalität
 Icelandic: fjöldatala
 Italian: cardinalità , numerosità , potenza
 Norwegian: kardinalitet
 Portuguese: cardinalidade
 Spanish: cardinalidad
Extensive Definition
In mathematics, the cardinality
of a set is
a measure of the "number of elements
of the set". For example, the set A = contains 3 elements, and
therefore A has a cardinality of 3. There are two approaches to
cardinality – one which compares sets directly using
bijections and
injections,
and another which uses cardinal
numbers.
Comparing sets
Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B. For example, the set E = of positive even numbers has the same cardinality as the set N = of natural numbers, since the function f(n) = 2n is a bijection from N to E.A set A has cardinality greater than or equal to
the cardinality of B if there exists an injective function from B
into A. The set A has cardinality strictly greater than the
cardinality of B,if there is an injective function, but no
bijective function, from B to A. For example, the set R of all
real
numbers has cardinality strictly greater than the cardinality
of the set N of all natural numbers, because the inclusion map i :
N → R is injective, but it can be shown that there does not exist a
bijective function from N to R.
Cardinal numbers
Above, "cardinality" was defined functionally.
That is, the "cardinality" of a set was not defined as a specific
object itself. However, such an object can be defined as
follows.
The relation of having the same cardinality is
called equinumerosity, and this
is an equivalence
relation on the class
of all sets. The equivalence
class of a set A under this relation then consists of all those
sets which have the same cardinality as A. There are two ways to
define the "cardinality of a set":
 The cardinality of a set A is defined as its equivalence class under equinumerosity.
 A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.
Cardinality of set S
is denoted  S . Cardinality of its power set is
denoted 2 S . Cardinalities of the infinite
sets are denoted
 \aleph_0
The cardinality of the natural
numbers is denoted alephnull
(ℵ0), while the cardinality of the real numbers
is denoted c, and is also referred to as the
cardinality of the continuum.
Finite, countable and uncountable sets
If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions: Any set X with cardinality less than that of the natural numbers, or  X  <  N , is said to be a finite set.
 Any set X that has the same cardinality as the set of the natural numbers, or  X  =  N  = ℵ0, is said to be a countably infinite set.
 Any set X with cardinality greater than that of the natural numbers, or  X  >  N , for example  R  = c >  N , is said to be uncountable.
Infinite sets
Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel.Dedekind simply defined an infinite set as one
having the same size as at least one of its "proper"
parts; this notion of infinity is called Dedekind
infinite.
Cantor introduced the abovementioned cardinal
numbers, and showed that some infinite sets are greater than
others. The smallest infinite cardinality is that of the natural
numbers (ℵ0).
Cardinality of the continuum
One of Cantor's most important results was that
the
cardinality of the continuum (c) is greater than that of the
natural numbers (ℵ0); that is, there are more real
numbers R than whole numbers N. Namely, Cantor showed that
 \mathbf = 2^ >
 (see Cantor's diagonal argument).
The continuum
hypothesis states that there is no cardinal
number between the cardinality of the reals and the cardinality
of the natural numbers, that is,
 \mathbf = \aleph_1 = \beth_1
 (see Beth one).
Cardinal arithmetic can be used to show not only
that the number of points in a real
number line is equal to the number of points in any segment of
that line, but that this is equal to the number of points on a
plane and, indeed, in any finitedimensional space. These results
are highly counterintuitive, because they imply that there exist
proper
subsets and proper
supersets of an infinite set S that have the same size as S,
although S contains elements that do not belong to its subsets, and
the supersets of S contain elements that are not included in
it.
The first of these results is apparent by
considering, for instance, the tangent
function, which provides a onetoone
correspondence between the interval
(−½π, ½π) and R (see also
Hilbert's paradox of the Grand Hotel).
The second result was first demonstrated by
Cantor in 1878, but it became more apparent in 1890, when Giuseppe
Peano introduced the spacefilling
curves, curved lines that twist and turn enough to fill the
whole of any square, or cube, or hypercube, or
finitedimensional space. These curves are not a direct proof that
a line has the same number of points as a finitedimensional space,
but they can be easily used to obtain
such a proof.
Cantor also showed that sets with cardinality
strictly greater than \mathbf c exist (see his
generalized diagonal argument and theorem).
They include, for instance:

 the set of all subsets of R, i.e., the power set of R, written P(R) or 2R
 the set RR of all functions from R to R
Both have cardinality
 2^\mathbf = \beth_2 > \mathbf c
 (see Beth two).
The
cardinal equalities \mathbf^2 = \mathbf, \mathbf c^ = \mathbf
c, and \mathbf c ^ = 2^ can be demonstrated using cardinal
arithmetic:
 \mathbf^2 = \left(2^\right)^2 = 2^ = 2^ = \mathbf,
 \mathbf c^ = \left(2^\right)^ = 2^ = 2^ = \mathbf,
 \mathbf c ^ = \left(2^\right)^ = 2^ = 2^.
Examples and properties
 If X = and Y = , then  X  =  Y  because is a bijection between the sets X and Y. The cardinality of each of X and Y is 3.
 If  X  <  Y , then there exists Z such that  X  =  Z  and Z ⊆ Y.
See also
cardinality in Bulgarian: Мощност на
множество
cardinality in Czech: Mohutnost
cardinality in Danish: Kardinalitet
cardinality in German: Mächtigkeit
(Mathematik)
cardinality in Korean: 기수
cardinality in Icelandic: Fjöldatala
cardinality in Italian: Cardinalità
cardinality in Hungarian: Számosság
cardinality in Norwegian: Kardinalitet
cardinality in Portuguese: Cardinalidade
cardinality in Romanian: Cardinal
(matematică)
cardinality in Russian: Мощность множества
cardinality in Finnish: Mahtavuus
cardinality in Swedish: Kardinalitet
cardinality in Ukrainian: MANZ
cardinality in Ukrainian: Потужність
множини
cardinality in Chinese: 勢