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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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. Beginning in the late 19th century, this concept was generalized to
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch ...
s, which allows one to distinguish between the different types of infinity, and to perform
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
on them. There are two approaches to cardinality: one which compares sets directly using
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
s and
injection Injection or injected may refer to: Science and technology * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

injection
s, and another which uses
cardinal number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a
vertical bar The vertical bar, , is a glyph The term glyph is used in typography File:metal movable type.jpg, 225px, Movable type being assembled on a composing stick using pieces that are stored in the type case shown below it Typography is the ...
on each side; this is the same notation as
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
, and the meaning depends on
context Context may refer to: * Context (language use) In semiotics, linguistics, sociology and anthropology, context refers to those objects or entities which surround a ''focal event'', in these disciplines typically a communication, communicative event ...
. The cardinality of a set A may alternatively be denoted by n(A), A, \operatorname(A), or \#A.


Comparing sets

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).


Definition 1: =

:Two sets ''A'' and ''B'' have the same cardinality if there exists a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
(a.k.a., one-to-one correspondence) from ''A'' to ''B'', that is, a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from ''A'' to ''B'' that is both
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Such sets are said to be ''equipotent'', ''equipollent'', or ''
equinumerous In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
''. This relationship can also be denoted ''A'' ≈ ''B'' or ''A'' ~ ''B''. :For example, the set ''E'' = of non-negative
even number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s has the same cardinality as the set N = of
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

natural numbers
, since the function ''f''(''n'') = 2''n'' is a bijection from N to ''E'' (see picture).


Definition 2: ≤

:''A'' has cardinality less than or equal to the cardinality of ''B'', if there exists an injective function from ''A'' into ''B''.


Definition 3: <

:''A'' has cardinality strictly less than the cardinality of ''B'', if there is an injective function, but no bijective function, from ''A'' to ''B''. :For example, the set N of all
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

natural numbers
has cardinality strictly less than its
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
''P''(N), because ''g''(''n'') = is an injective function from N to ''P''(N), and it can be shown that no function from N to ''P''(N) can be bijective (see picture). By a similar argument, N has cardinality strictly less than the cardinality of the set R of all
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. For proofs, see
Cantor's diagonal argument In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a bran ...
or
Cantor's first uncountability proof Cantor's first set theory article contains Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor e ...
. If ≤ and ≤ , then = (a fact known as
Schröder–Bernstein theoremIn set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the Set (mathematics), sets and , then there exists a bijection, bijective function . In terms of the cardinality of the two sets, this c ...
). The
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
is equivalent to the statement that ≤ or ≤ for every ''A'', ''B''.


Cardinal numbers

In the above section, "cardinality" of a set was defined functionally. In other words, it 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and this is an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
on the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of all sets. The
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
. Assuming the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
, the cardinalities of the
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch ...
s are denoted :\aleph_0 < \aleph_1 < \aleph_2 < \ldots . For each
ordinal Ordinal may refer to: * Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale * Ordinal date, a simple form of expressing a date using only the year and the day number within that year * O ...
\alpha, \aleph_ is the least cardinal number greater than \aleph_\alpha. The cardinality of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s is denoted
aleph-null In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(\aleph_0), while the cardinality of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is denoted by "\mathfrak c" (a lowercase fraktur script "c"), and is also referred to as the
cardinality of the continuum In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
. Cantor showed, using the
diagonal argumentDiagonal argument in mathematics may refer to: *Cantor's diagonal argument (the earliest) *Cantor's theorem *Halting problem *Diagonal lemma See also

* Diagonalization (disambiguation) {{mathdab ...
, that >\aleph_0. We can show that \mathfrak c = 2^, this also being the cardinality of the set of all subsets of the natural numbers. The
continuum hypothesis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
says that \aleph_1 = 2^, i.e. 2^ is the smallest cardinal number bigger than \aleph_0, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independen ...
of ZFC, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent. For more detail, see § Cardinality of the continuum below.


Finite, countable and uncountable sets

If the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, or ,  ''X'' , < ,  N , , is said to be a
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. *Any set ''X'' that has the same cardinality as the set of the natural numbers, or ,  ''X'' , = ,  N , = \aleph_0, is said to be a
countably infinite In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
set. *Any set ''X'' with cardinality greater than that of the natural numbers, or ,  ''X'' , > ,  N , , for example ,  R , = \mathfrak c > ,  N , , is said to be
uncountable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


Infinite sets

Our intuition gained from
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s breaks down when dealing with
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch ...
s. In the late nineteenth century
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...
,
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citiz ...
and others rejected the view that the whole cannot be the same size as the part.
Reprinted in: Here: p.413 bottom
One example of this is Hilbert's paradox of the Grand Hotel. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (\aleph_0).


Cardinality of the continuum

One of Cantor's most important results was that the
cardinality of the continuum In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
(\mathfrak) is greater than that of the natural numbers (\aleph_0); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that \mathfrak = 2^ = \beth_1 (see Beth one) satisfies: :2^ > \aleph_0 :(see
Cantor's diagonal argument In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a bran ...
or
Cantor's first uncountability proof Cantor's first set theory article contains Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor e ...
). The
continuum hypothesis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
states that there is no
cardinal number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
between the cardinality of the reals and the cardinality of the natural numbers, that is, :2^ = \aleph_1 However, this hypothesis can neither be proved nor disproved within the widely accepted
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, if ZFC is consistent. Cardinal arithmetic can be used to show not only that the number of points in a
real number line Real may refer to: Currencies * Brazilian real The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil ...

real number line
is equal to the number of points in any
segment Segment or segmentation may refer to: Biology *Segmentation (biology), the division of body plans into a series of repetitive segments **Segmentation in the human nervous system *Internodal segment, the portion of a nerve fiber between two Nodes of ...

segment
of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist
proper subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s 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 In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
, which provides a
one-to-one correspondence In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...
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 Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Giuseppe Peano
introduced the
space-filling curve In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
s, curved lines that twist and turn enough to fill the whole of any square, or cube, or
hypercube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypercube
, or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof. Cantor also showed that sets with cardinality strictly greater than \mathfrak c exist (see his generalized diagonal argument and
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
). They include, for instance: :* the set of all subsets of R, i.e., the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of R, written ''P''(R) or 2R :* the set RR of all functions from R to R Both have cardinality :2^\mathfrak = \beth_2 > \mathfrak c :(see Beth two). The cardinal equalities \mathfrak^2 = \mathfrak, \mathfrak c^ = \mathfrak c, and \mathfrak c ^ = 2^ can be demonstrated using
cardinal arithmetic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: :\mathfrak^2 = \left(2^\right)^2 = 2^ = 2^ = \mathfrak, :\mathfrak c^ = \left(2^\right)^ = 2^ = 2^ = \mathfrak, : \mathfrak 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''. *If ,  ''X'' , ≤ ,  ''Y'' , and ,  ''Y'' , ≤ ,  ''X'' , , then ,  ''X'' , = ,  ''Y'' , . This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem. * Sets with cardinality of the continuum include the set of all real numbers, the set of all
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s and the interval
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math>.


Union and intersection

If ''A'' and ''B'' are
disjoint sets In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

disjoint sets
, then :\left\vert A \cup B \right\vert = \left\vert A \right\vert + \left\vert B \right\vert. From this, one can show that in general, the cardinalities of unions and intersections are related by the following equation:Applied Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983, (student edition), (library edition) : \left\vert C \cup D \right\vert + \left\vert C \cap D \right\vert = \left\vert C \right\vert + \left\vert D \right\vert.


See also

*
Aleph number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Beth number In mathematics, the beth numbers are a certain sequence of infinite set, infinite cardinal numbers, conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second Hebrew alphabet, Hebrew letter (bet (letter), beth). ...
*
Cantor's paradox In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
*
Cantor's theorem In elementary set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. ...
*
Countable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
* Ordinality *
Pigeonhole principle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...


References

{{Authority control Basic concepts in infinite set theory