In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the cardinality of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is a measure of the number of
elements of the set. For example, the set
contains 3 elements, and therefore
has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to
infinite sets, which allows one to distinguish between different types of infinity, and to perform
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
on them. There are two approaches to cardinality: one which compares sets directly using
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
s and
injections, and another which uses
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
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
is usually denoted
, with a
vertical bar on each side; this is the same notation as
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, and the meaning depends on
context
Context may refer to:
* Context (language use), the relevant constraints of the communicative situation that influence language use, language variation, and discourse summary
Computing
* Context (computing), the virtual environment required to s ...
. The cardinality of a set
may alternatively be denoted by
,
,
, or
.
History
A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago. Human expression of cardinality is seen as early as years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells. The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and the manipulation of numbers without reference to a specific group of things or events.
From the 6th century BCE, the writings of Greek philosophers show the first hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing.
The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. In Euclid's ''
Elements'',
commensurability was described as the ability to compare the length of two line segments, ''a'' and ''b'', as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into both ''a'' and ''b''. But with the discovery of
irrational numbers, it was seen that even the infinite set of all rational numbers was not enough to describe the length of every possible line segment. Still, there was no concept of infinite sets as something that had cardinality.
To better understand infinite sets, a notion of cardinality was formulated circa 1880 by
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
, the originator of
set theory
Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
. He examined the process of equating two sets with
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, a one-to-one correspondence between the elements of two sets based on a unique relationship. In 1891, with the publication of
Cantor's diagonal argument, he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e.
uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
s that contain more elements than there are in the infinite set of natural numbers.
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, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
(a.k.a., one-to-one correspondence) from ''A'' to ''B'',
that is, a
function from ''A'' to ''B'' that is both
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
and
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
. Such sets are said to be ''equipotent'', ''equipollent'', or ''
equinumerous''. This relationship can also be denoted ''A'' ≈ ''B'' or ''A'' ~ ''B''.
:For example, the set ''E'' = of non-negative
even numbers 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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, since the function ''f''(''n'') = 2''n'' is a bijection from N to ''E'' (see picture).
:For finite sets ''A'' and ''B'', if ''some'' bijection exists from ''A'' to ''B'', then ''each'' injective or surjective function from ''A'' to ''B'' is a bijection. This is no longer true for infinite ''A'' and ''B''. For example, the function ''g'' from N to ''E'', defined by ''g''(''n'') = 4''n'' is injective, but not surjective, and ''h'' from N to ''E'', defined by ''h''(''n'') = ''n'' - (''n''
mod 2) is surjective, but not injective. Neither ''g'' nor ''h'' can challenge = , which was established by the existence of ''f''.
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
has cardinality strictly less than its
power set ''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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. For proofs, see
Cantor's diagonal argument or
Cantor's first uncountability proof
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is ...
.
If ≤ and ≤ , then = (a fact known as
Schröder–Bernstein theorem). 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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, and this is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
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 ...
of all sets. The
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
in
axiomatic set theory
Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
.
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, the cardinalities of the
infinite sets are denoted
:
For each
ordinal ,
is the least cardinal number greater than
.
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s is denoted
aleph-null (
), while the cardinality of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is denoted by "
" (a lowercase
fraktur script "c"), and is also referred to as the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...
. Cantor showed, using the
diagonal argument, that
. We can show that
, this also being the cardinality of the set of all subsets of the natural numbers.
The
continuum hypothesis says that
, i.e.
is the smallest cardinal number bigger than
, 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
* Independe ...
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, 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 , =
, is said to be a
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
set.
*Any set ''X'' with cardinality greater than that of the natural numbers, or , ''X'' , > , N , , for example , R , =
> , 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
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
,
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
,
Richard Dedekind 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 (
).
Cardinality of the continuum
One of Cantor's most important results was that the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...
(
) is greater than that of the natural numbers (
); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that
(see
Beth one) satisfies:
:
:(see
Cantor's diagonal argument or
Cantor's first uncountability proof
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is ...
).
The
continuum hypothesis states that there is no
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
between the cardinality of the reals and the cardinality of the natural numbers, that is,
:
However, this hypothesis can neither be proved nor disproved within the widely accepted
ZFC axiomatic set theory
Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
, if ZFC is consistent.
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 finite-dimensional space. These results are highly counterintuitive, because they imply that there exist
proper subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
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 a ...
, which provides a
one-to-one 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
space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, ...
s, curved lines that twist and turn enough to fill the whole of any square, or cube, or
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
, 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
exist (see his
generalized diagonal argument and
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
). They include, for instance:
:* the set of all subsets of R, i.e., the
power set of R, written ''P''(R) or 2
R
:* the set R
R of all functions from R to R
Both have cardinality
:
:(see
Beth two).
The
cardinal equalities and
can be demonstrated using
cardinal arithmetic:
:
:
:
Examples and properties
* If ''X'' = and ''Y'' = , where ''a'', ''b'', and ''c'' are distinct, 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 numbers and the interval