Cardinality

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 using bijections and injections, and another which uses cardinal numbers.

The cardinality of a set "A" is denoted | "A" |, with a vertical bar on each side; this is the same notation as absolute 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 numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural 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 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.

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.

The cardinalities of the infinite sets are denoted :aleph_0 < aleph_1 < aleph_2 < ldots . For each ordinal α, IPA|&alefsym;α + 1 is the least cardinal number greater than IPA|&alefsym;α.

The cardinality of the natural numbers is denoted aleph-null (IPA|&alefsym;0), while the cardinality of the real numbers is denoted 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, or |&thinsp;"X"&thinsp;| < |&thinsp;N&thinsp;|, is said to be a finite set.
*Any set "X" that has the same cardinality as the set of the natural numbers, or |&thinsp;"X"&thinsp;| = |&thinsp;N&thinsp;| = IPA|&alefsym;0, is said to be a countably infinite set.
*Any set "X" with cardinality greater than that of the natural numbers, or |&thinsp;"X"&thinsp;| > |&thinsp;N&thinsp;|, for example |&thinsp;R&thinsp;| = c > |&thinsp;N&thinsp;|, 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 (IPA|&alefsym;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|&alefsym;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 = eth_1:(see Beth one).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 (−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 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.

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., the power set of R, written "P"(R) or 2R:* the set RR 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 |&thinsp;"X"&thinsp;| = |&thinsp;"Y"&thinsp;| 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 |&thinsp;"X"&thinsp;| &lt; |&thinsp;"Y"&thinsp;|, then there exists "Z" such that |&thinsp;"X"&thinsp;| = |&thinsp;"Z"&thinsp;| and "Z" ⊆ "Y".

* Sets with cardinality c

ee also

* Aleph number
* Beth number


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • cardinality — 1520s, condition of being a cardinal, from CARDINAL (Cf. cardinal) (q.v.) + ITY (Cf. ity). Mathematical sense is from 1935 …   Etymology dictionary

  • cardinality — noun a) Of a set, the number of elements it contains. The empty set has a cardinality of zero. b) The property of a relationship between a database table and another one, specifying whether it is one to one, one to many, many to one, or many to… …   Wiktionary

  • cardinality — The cardinality of a set is the cardinal number that measures the number of its members …   Philosophy dictionary

  • Cardinality (disambiguation) — Cardinality may refer to* Cardinality of a set, a measure of the number of elements of a set in mathematics * Cardinality (data modeling), a term in database design * Cardinality (SQL statements), a term used in SQL statements * Cardinal utility …   Wikipedia

  • Cardinality (SQL statements) — In SQL (Structured Query Language), the term cardinality refers to the uniqueness of data values contained in a particular column (attribute) of a database table. The lower the cardinality, the more duplicated elements in a column. Thus, a column …   Wikipedia

  • Cardinality of the continuum — In mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size (cardinality) of the set of real numbers mathbb R (sometimes called the continuum). The cardinality of mathbb R is often denoted by… …   Wikipedia

  • Cardinality (data modeling) — In data modeling, the cardinality of one data table with respect to another data table is a critical aspect of database design. Relationships between data tables define cardinality when explaining how each table links to another. In the… …   Wikipedia

  • Cardinality equals variety — The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the same distance in pitch class space. In general,… …   Wikipedia

  • cardinality — noun (plural ties) Etymology: 1cardinal + ity Date: 1935 the number of elements in a given mathematical set …   New Collegiate Dictionary

  • cardinality — /kahr dn al i tee/, n., pl. cardinalities. Math. (of a set) the cardinal number indicating the number of elements in the set. [1930 35; CARDINAL + ITY] * * * …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”