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Encyclopedia > Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about sets in mathematics. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... A bijective function. ... An injective function. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...

## Contents

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 = {2, 4, 6, ...} of positive even numbers has the same cardinality as the set N = {1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. A bijective function. ... An injective function. ... A surjective function. ... Partial plot of a function f. ... A negative number is a number that is less than zero, such as âˆ’3. ... In mathematics, any integer (whole number) is either even or odd. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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 A has cardinality greater than or equal to the cardinality of B, but A and B do not have the same cardinality. In other words, if there is an injective function from B to A, 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 : NR is injective, but it can be shown that there does not exist a bijective function from N to R. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

## Countable and uncountable sets

If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions: In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Generally, a trichotomy is a splitting into three disjoint parts. ...

• Any set with cardinality less than that of the natural numbers is said to be a finite set.
• Any set that has the same cardinality as the set of the natural numbers is said to be a countably infinite set.
• Any set with cardinality greater than that of the natural numbers is said to be uncountable.

In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a countable set is a set with the same cardinality (i. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ...

## Cardinal numbers

Main article: Cardinal number

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. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...

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": Two sets A and B are said to be equinumerous if they have the same cardinality, i. ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

1. The cardinality of a set A is defined as its equivalence class under equinumerosity.
2. 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 | . The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... This article or section is in need of attention from an expert on the subject. ... This article is about sets in mathematics. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

Cardinalities of the infinite sets are denoted $aleph_0 < aleph_1 < aleph_2 < ... .$ For each ordinal α, $aleph_{alpha+1}$ is the least cardinal number greater than $aleph_alpha.$ Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

The cardinality of the natural numbers is denoted aleph-null (${aleph_0}$), while the cardinality of the real numbers is denoted $mathbf{c}$. It can be shown that $mathbf{c} = 2^{aleph_0} > {aleph_0}$ (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, $mathbf{c} = aleph_1$. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... Please refer to Real vs. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...

## Examples and properties

• If X = {a, b, c} and Y = {apples, oranges, peaches}, then | X | = | Y | , because ${ langle a, apples rangle, langle b, oranges rangle, langle c, peaches rangle }$ is a bijection between them. Their cardinality is 3.
• If $|X| leq |Y|$, then there exists Z such that | X | = | Z | and $Z subseteq Y.$
• There is no largest cardinal number. That is, given any set X, there is a set Y such that X < Y. In particular, Y may be taken to be the powerset of X.

In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, an uncountable set is a set which is not countable. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ...

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