Cardinality

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In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {1, 2, 3} 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 = {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 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, 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 $2^$

Cardinalities of the infinite sets are denoted $aleph_0 < aleph_1 < aleph_2 < ... .$ For each ordinal $alpha$ , $aleph_{alpha+1}$ is the least cardinal number greater than $aleph_alpha.$

The cardinality of the natural numbers is denoted aleph-null ( ${aleph_0}$ ), while the cardinality of the real numbers is denoted $mathbf{c}$ , 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 (|X| < |N|) is said to be a finite set.
Any set X that has the same cardinality as the set of the natural numbers (|X| = |N| = $aleph_0$ ) is said to be a countably infinite set.
Any set X with cardinality greater than that of the natural numbers (|X| > |N|, for example |R| = $mathbf{c}$ > |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 above-mentioned cardinal numbers, and showed 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 ( $mathbf c$ ) is greater than that of the natural numbers ( ${aleph_0}$ ); that is, there are more real numbers R than whole numbers N. Namely, Cantor showed 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, $mathbf{c} = aleph_1$ . However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory, 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 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 one-to-one correspondence between the interval [-0.5π, 0.5π] 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 curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or 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 easily used to obtain such a proof.

The cardinal equality $c^2=c$ can be demonstrated using cardinal arithmetic: $mathbf{c}^2 = (2^{aleph_0})^2 = 2^{2times{aleph_0}} = 2^{aleph_0} = mathbf{c}.$ This argument is a condensed version of the notion of interleaving two binary sequences: let $0.a_0a_1a_2ldots$ be the binary expansion of $x$ and let $0.b_0b_1b_2ldots$ be the binary expansion of $y$ . Then $z=0.a_0b_0a_1b_1ldots$ , the interleaving of the binary expansions, is a well-defined function when $x$ and $y$ have unique binary expansions. Only countably many reals have non-unique binary expansions.

Cantor's generalized diagonal argument shows that $mathbf c < P(mathbf c)$ which implies $mathbf c < 2^{mathbf c} leq mathbf c^{mathbf c}$ . Here $P(mathbf c)equiv 2^{mathbf c}$ denotes the power set of $mathbf c$ , the set of all subsets of $mathbf c$ , and $mathbf c^{mathbf c}$ denotes the set of functions from $mathbf c$ to $mathbf c$ . Furthermore $mathbf c ^{mathbf c} = (2^{aleph_0})^{mathbf c} = 2^{mathbf ctimesaleph_0} = 2^{mathbf c} = beth_2$ . See Beth number#Beth two.

Examples and properties

If $X$ = {a, b, c} and $Y$ = {apples, oranges, peaches}, then $|X| = |Y|,$ because ${ langle a, mbox{apples} rangle, langle b, mbox{oranges} rangle, langle c, mbox{peaches} rangle }$ is a bijection between them. Their cardinality is 3.

If $|X| leq |Y|$ , then there exists $Z ,$ such that $|X| = |Z|,$ , and $Z subseteq Y.$

Sets with cardinality $mathbf c$