- Cardinality
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 usingbijection s and injections, and another which usescardinal number s.The cardinality of a set "A" is denoted | "A" |, with a

vertical bar on each side; this is the same notation asabsolute value and the meaning depends on context.**Comparing sets**

= Case 1: | "A" | = | "B" | =: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" = {0, 2, 4, 6, ...} of non-negative

even number s has the same cardinality as the set**N**= {0, 1, 2, 3, ...} ofnatural numbers , since the function "f"("n") = 2"n" is a bijection from**N**to "E".**Case 2: | "A" | ≥ | "B" |**:"A" has cardinality greater than or equal to the cardinality of "B" if there exists an injective function from "B" into "A".

**Case 3: | "A" | > | "B" |**:"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 allreal number s 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 anequivalence relation on the class of all sets. Theequivalence 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 ofcardinal number inaxiomatic set theory .The cardinalities of the infinite sets are denoted :$aleph\_0\; <\; aleph\_1\; <\; aleph\_2\; <\; ldots\; .$ For each

ordinal α, IPA|ℵ_{α + 1}is the least cardinal number greater than IPA|ℵ_{α}.The cardinality of the

natural numbers is denoted aleph-null (IPA|ℵ_{0}), while the cardinality of thereal numbers is denoted**c**, and is also referred to as thecardinality 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 number s, or | "X" | < | **N** |, is said to be afinite set .

*Any set "X" that has the same cardinality as the set of the natural numbers, or | "X" | = | **N** | = IPA|ℵ_{0}, 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 set s breaks down when dealing withinfinite set s. In the late nineteenth centuryGeorg Cantor ,Gottlob Frege ,Richard Dedekind and others rejected the view of Galileo (which derived fromEuclid ) that the whole cannot be the same size as the part. One example of this isHilbert'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 (IPA|ℵ

_{0}).**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 (IPA|ℵ_{0}); that is, there are more real numbers**R**than whole numbers**N**. Namely, Cantor showed that :$mathbf\{c\}\; =\; 2^\{aleph\_0\}\; >\; \{aleph\_0\}$ :(seeCantor's diagonal argument ).The

continuum hypothesis states that there is nocardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, :$mathbf\{c\}\; =\; aleph\_1\; =\; eth\_1$:(see Beth one).However, this hypothesis can neither be proved nor disproved within the widely acceptedZFC 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 existproper subset s andproper superset s 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 aone-to-one correspondence between theinterval (−½π, ½π) and**R**(see alsoHilbert'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 thespace-filling curve s, curved lines that twist and turn enough to fill the whole of any square, or cube, orhypercube , 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.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., thepower set of**R**, written "P"(**R**) or 2^{R}:* the set**R**^{R}of all functions from**R**to**R**Both have cardinality :$2^mathbf\; \{c\}\; =\; eth\_2\; mathbf\; c$ :(see Beth two).

The cardinal equalities $mathbf\{c\}^2\; =\; mathbf\{c\},$ $mathbf\; c^\{aleph\_0\}\; =\; mathbf\; c,$ and $mathbf\; c\; ^\{mathbf\; c\}\; =\; 2^\{mathbf\; c\}$ can be demonstrated using

cardinal arithmetic ::$mathbf\{c\}^2\; =\; left(2^\{aleph\_0\}\; ight)^2\; =\; 2^\{2\; imes\{aleph\_0\; =\; 2^\{aleph\_0\}\; =\; mathbf\{c\},$:$mathbf\; c^\{aleph\_0\}\; =\; left(2^\{aleph\_0\}\; ight)^\{aleph\_0\}\; =\; 2^$aleph_0} imes{aleph_0 = 2^{aleph_0} = mathbf{c},:$mathbf\; c\; ^\{mathbf\; c\}\; =\; left(2^\{aleph\_0\}\; ight)^\{mathbf\; c\}\; =\; 2^\{mathbf\; c\; imesaleph\_0\}\; =\; 2^\{mathbf\; c\}.$**Examples and properties*** If "X" = {"a", "b", "c"} and "Y" = {apples, oranges, peaches}, then | "X" | = | "Y" | because {("a", apples), ("b", oranges), ("c", peaches)} 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".**ee also***

Aleph number

*Beth number

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