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In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometric shapes, variables, or even other sets. A set may be finite or infinite, depending whether the number of its elements is finite or not. There is a unique set with no elements, called the empty set; a set with a single element is a singleton.
Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.
ContextEdit
Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialTemplate:Mdashmeaning that it is the result of an endless processTemplate:Mdashand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specifically, a line was not considered as the set of its points, but as a locus where points may be located.
The mathematical study of infinite sets began with Georg Cantor (1845–1918). This provided some counterintuitive facts and paradoxes. For example, the number line has an infinite number of elements that is strictly larger than the infinite number of natural numbers, and any line segment has the same number of elements as the whole space. Also, Russell's paradox implies that the phrase "the set of all sets" is self-contradictory.
Together with other counterintuitive results, this led to the foundational crisis of mathematics, which was eventually resolved with the general adoption of Zermelo–Fraenkel set theory as a robust foundation of set theory and all mathematics.
Meanwhile, sets started to be widely used in all mathematics. In particular, algebraic structures and mathematical spaces are typically defined in terms of sets. Also, many older mathematical results are restated in terms of sets. For example, Euclid's theorem is often stated as "the set of the prime numbers is infinite". This wide use of sets in mathematics was prophesied by David Hilbert when saying: "No one will drive us from the paradise which Cantor created for us."<ref>Template:Citation
- "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können."
- Translated in Template:Citation</ref>
Generally, the common usage of sets in mathematics does not require the full power of Zermelo–Fraenkel set theory. In mathematical practice, sets can be manipulated independently of the logical framework of this theory.
The object of this article is to summarize the manipulation rules and properties of sets that are commonly used in mathematics, without reference to any logical framework. For the branch of mathematics that studies sets, see Set theory; for an informal presentation of the corresponding logical framework, see Naive set theory; for a more formal presentation, see Axiomatic set theory and Zermelo–Fraenkel set theory.
Basic notionsEdit
In mathematics, a set is a collection of different things.<ref name="Cantor">Template:Cite book Here: p.85</ref><ref name="JainAhmad1995">Template:Cite book</ref><ref name="Goldberg1986">Template:Cite book</ref><ref name="CormenCormen2001">Template:Cite book</ref> These things are called elements or members of the set and are typically mathematical objects of any kind such as numbers, symbols, points in space, lines, other geometrical shapes, variables, functions, or even other sets.Template:Sfn<ref>Template:Cite book</ref> A set may also be called a collection or family, especially when its elements are themselves sets; this may avoid the confusion between the set and its members, and may make reading easier. A set may be specified either by listing its elements or by a property that characterizes its elements, such as for the set of the prime numbers or the set of all students in a given class.<ref name=":0">Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
If Template:Tmath is an element of a set Template:Tmath, one says that Template:Tmath belongs to Template:Tmath or is in Template:Tmath, and this is written as Template:Tmath.Template:Sfn The statement "Template:Tmath is not in Template:Tmath" is written as Template:Tmath, which can also be read as "y is not in B".<ref name="CapinskiKopp2004">Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> For example, if Template:Tmath is the set of the integers, one has Template:Tmath and Template:Tmath. Each set is uniquely characterized by its elements. In particular, two sets that have precisely the same elements are equal (they are the same set).<ref name="Stoll">Template:Cite book</ref> This property, called extensionality, can be written in formula as <math display="block">A=B \iff \forall x\; (x\in A \iff x \in B).</math>This implies that there is only one set with no element, the empty set (or null set) that is denoted Template:Tmath,Template:Efn or Template:TmathTemplate:Sfn<ref name="LeungChen1992">Template:Cite book</ref> A singleton is a set with exactly one element.Template:Efn If Template:Tmath is this element, the singleton is denoted Template:Tmath If Template:Tmath is itself a set, it must not be confused with Template:Tmath For example, Template:Tmath is a set with no elements, while Template:Tmath is a singleton with Template:Tmath as its unique element.
A set is finite if there exists a natural number Template:Tmath such that the Template:Tmath first natural numbers can be put in one to one correspondence with the elements of the set. In this case, one says that Template:Tmath is the number of elements of the set. A set is infinite if such an Template:Tmath does not exist. The empty set is a finite set with Template:Tmath elements.
The natural numbers form an infinite set, commonly denoted Template:Tmath. Other examples of infinite sets include number sets that contain the natural numbers, real vector spaces, curves and most sorts of spaces.
Specifying a setEdit
Extensionality implies that for specifying a set, one has either to list its elements or to provide a property that uniquely characterizes the set elements.
Roster notationEdit
Roster or enumeration notation is a notation introduced by Ernst Zermelo in 1908 that specifies a set by listing its elements between braces, separated by commas.<ref>A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair", p.278. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</ref><ref name="Roberts2009">Template:Cite book</ref><ref name="JohnsonJohnson2004">Template:Cite book</ref><ref name="BelloKaul2013">Template:Cite book</ref><ref name="Epp2010">Template:Cite book</ref> For example, one knows that <math>\{4, 2, 1, 3\}</math> and <math>\{\text{blue, white, red}\}</math> denote sets and not tuples because of the enclosing braces.
Above notations Template:Tmath and Template:Tmath for the empty set and for a singleton are examples of roster notation.
When specifying sets, it only matters whether each distinct element is in the set or not; this means a set does not change if elements are repeated or arranged in a different order. For example,<ref>Template:Cite book</ref><ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="DalenDoets2014">Template:Cite book</ref>
<math display =block>\{1,2,3,4\}=\{4, 2, 1, 3\} = \{4, 2, 4, 3, 1, 3\}.</math>
When there is a clear pattern for generating all set elements, one can use ellipses for abbreviating the notation,<ref name="BastaDeLong2013">Template:Cite book</ref><ref name="BrackenMiller2013">Template:Cite book</ref> such as in <math display =block>\{1,2,3,\ldots,1000\}</math> for the positive integers not greater than Template:Tmath.
Ellipses allow also expanding roster notation to some infinite sets. For example, the set of all integers can be denoted as
<math display =block>\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}</math>
or
<math display =block>\{0, 1, -1, 2, -2, 3, -3, \ldots\}.</math>
Set-builder notationEdit
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Set-builder notation specifies a set as being the set of all elements that satisfy some logical formula.<ref name="Ruda2011">Template:Cite book</ref><ref name="Lucas1990">Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> More precisely, if Template:Tmath is a logical formula depending on a variable Template:Tmath, which evaluates to true or false depending on the value of Template:Tmath, then <math display=block>\{x \mid P(x)\}</math> or<ref name="Steinlage1987">Template:Cite book</ref> <math display=block>\{x : P(x)\}</math> denotes the set of all Template:Tmath for which Template:Tmath is true.<ref name=":0" /> For example, a set Template:Mvar can be specified as follows: <math display="block">F = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math> In this notation, the vertical bar "|" is read as "such that", and the whole formula can be read as "Template:Mvar is the set of all Template:Mvar such that Template:Mvar is an integer in the range from 0 to 19 inclusive".
Some logical formulas, such as Template:Tmath or Template:Tmath cannot be used in set-builder notation because there is no set for which the elements are characterized by the formula. There are several ways for avoiding the problem. One may prove that the formula defines a set; this is often almost immediate, but may be very difficult.
One may also introduce a larger set Template:Tmath that must contain all elements of the specified set, and write the notation as <math display=block>\{x\mid x\in U \text{ and ...}\}</math> or <math display=block>\{x\in U\mid \text{ ...}\}.</math>
One may also define Template:Tmath once for all and take the convention that every variable that appears on the left of the vertical bar of the notation represents an element of Template:Tmath. This amounts to say that Template:Tmath is implicit in set-builder notation. In this case, Template:Tmath is often called the domain of discourse or a universe.
For example, with the convention that a lower case Latin letter may represent a real number and nothing else, the expression <math display=block>\{x\mid x\not\in \Q\}</math> is an abbreviation of <math display="block">\{x\in \R \mid x\not\in \Q\},</math> which defines the irrational numbers.
SubsetsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A subset of a set Template:Tmath is a set Template:Tmath such that every element of Template:Tmath is also an element of Template:Tmath.<ref name="Hausdorff2005">Template:Cite book</ref> If Template:Tmath is a subset of Template:Tmath, one says commonly that Template:Tmath is contained in Template:Tmath, Template:Tmath contains Template:Tmath, or Template:Tmath is a superset of Template:Tmath. This denoted Template:Tmath and Template:Tmath. However many authors use Template:Tmath and Template:Tmath instead. The definition of a subset can be expressed in notation as <math display=block>A \subseteq B \quad \text{if and only if}\quad \forall x\; (x\in A \implies x\in B).</math>
A set Template:Tmath is a proper subset of a set Template:Tmath if Template:Tmath and Template:Tmath. This is denoted Template:Tmath and Template:Tmath. When Template:Tmath is used for the subset relation, or in case of possible ambiguity, one uses commonly Template:Tmath and Template:Tmath.Template:Sfn
The relationship between sets established by ⊆ is called inclusion or containment. Equality between sets can be expressed in terms of subsets. Two sets are equal if and only if they contain each other: that is, Template:Math and Template:Math is equivalent to A = B.<ref name="Lucas1990"/><ref name=":0" /> The empty set is a subset of every set: Template:Math.Template:Sfn
Examples:
- The set of all humans is a proper subset of the set of all mammals.
- Template:Math.
- Template:Math
Basic operationsEdit
There are several standard operations that produce new sets from given sets, in the same way as addition and multiplication produce new numbers from given numbers. The operations that are considered in this section are those such that all elements of the produced sets belong to a previously defined set. These operations are commonly illustrated with Euler diagrams and Venn diagrams.<ref>Template:Cite book</ref>
The main basic operations on sets are the following ones.
IntersectionEdit
The intersection of two sets Template:Tmath and Template:Tmath is a set denoted Template:Tmath whose elements are those elements that belong to both Template:Tmath and Template:Tmath. That is, <math display=block>A \cap B=\{x\mid x\in A \land x\in B\},</math> where Template:Tmath denotes the logical and.
Intersection is associative and commutative; this means that for proceeding a sequence of intersections, one may proceed in any order, without the need of parentheses for specifying the order of operations. Intersection has no general identity element. However, if one restricts intersection to the subsets of a given set Template:Tmath, intersection has Template:Tmath as identity element.
If Template:Tmath is a nonempty set of sets, its intersection, denoted <math display=inline>\bigcap_{A\in \mathcal S} A,</math> is the set whose elements are those elements that belong to all sets in Template:Tmath. That is, <math display=block>\bigcap_{A\in \mathcal S} A =\{x\mid (\forall A\in \mathcal S)\; x\in A\}.</math>
These two definitions of the intersection coincide when Template:Tmath has two elements.
UnionEdit
The union of two sets Template:Tmath and Template:Tmath is a set denoted Template:Tmath whose elements are those elements that belong to Template:Tmath or Template:Tmath or both. That is, <math display=block>A \cup B=\{x\mid x\in A \lor x\in B\},</math> where Template:Tmath denotes the logical or.
Union is associative and commutative; this means that for proceeding a sequence of intersections, one may proceed in any order, without the need of parentheses for specifying the order of operations. The empty set is an identity element for the union operation.
If Template:Tmath is a set of sets, its union, denoted <math display=inline>\bigcup_{A\in \mathcal S} A,</math> is the set whose elements are those elements that belong to at least one set in Template:Tmath. That is, <math display=block>\bigcup_{A\in \mathcal S} A =\{x\mid (\exists A\in \mathcal S)\; x\in A\}.</math>
These two definitions of the union coincide when Template:Tmath has two elements.
Set differenceEdit
The set difference of two sets Template:Tmath and Template:Tmath, is a set, denoted Template:Tmath or Template:Tmath, whose elements are those elements that belong to Template:Tmath, but not to Template:Tmath. That is, <math display=block>A \setminus B=\{x\mid x\in A \land x\not\in B\},</math> where Template:Tmath denotes the logical and.
When Template:Tmath the difference Template:Tmath is also called the complement of Template:Tmath in Template:Tmath. When all sets that are considered are subsets of a fixed universal set Template:Tmath, the complement Template:Tmath is often called the absolute complement of Template:Tmath.
The symmetric difference of two sets Template:Tmath and Template:Tmath, denoted Template:Tmath, is the set of those elements that belong to Template:Mvar or Template:Mvar but not to both: <math display =block>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
Algebra of subsetsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The set of all subsets of a set Template:Tmath is called the powerset of Template:Tmath, often denoted Template:Tmath. The powerset is an algebraic structure whose main operations are union, intersection, set difference, symmetric difference and absolute complement (complement in Template:Tmath).
The powerset is a Boolean ring that has the symmetric difference as addition, the intersection as multiplication, the empty set as additive identity, Template:Tmath as multiplicative identity, and complement as additive inverse.
The powerset is also a Boolean algebra for which the join Template:Tmath is the union Template:Tmath, the meet Template:Tmath is the intersection Template:Tmath, and the negation is the set complement.
As every Boolean algebra, the power set is also a partially ordered set for set inclusion. It is also a complete lattice.
The axioms of these structures induce many identities relating subsets, which are detailed in the linked articles.
FunctionsEdit
Template:Main article A function from a set Template:MvarTemplate:Mdashthe domainTemplate:Mdashto a set Template:MvarTemplate:Mdashthe codomainTemplate:Mdashis a rule that assigns to each element of Template:Mvar a unique element of Template:Mvar. For example, the square function maps every real number Template:Mvar to Template:Math. Functions can be formally defined in terms of sets by means of their graph, which are subsets of the Cartesian product (see below) of the domain and the codomain.
Functions are fundamental for set theory, and examples are given in following sections.
Indexed familiesEdit
Intuitively, an indexed family is a set whose elements are labelled with the elements of another set, the index set. These labels allow the same element to occur several times in the family.
Formally, an indexed family is a function that has the index set as its domain. Generally, the usual functional notation Template:Tmath is not used for indexed families. Instead, the element of the index set is written as a subscript of the name of the family, such as in Template:Tmath.
When the index set is Template:Tmath, an indexed family is called an ordered pair. When the index set is the set of the Template:Tmath first natural numbers, an indexed family is called an Template:Tmath-tuple. When the index set is the set of all natural numbers an indexed family is called a sequence.
In all these cases, the natural order of the natural numbers allows omitting indices for explicit indexed families. For example, Template:Tmath denotes the 3-tuple Template:Tmath such that Template:Tmath.
The above notations <math display=inline>\bigcup_{A\in \mathcal S} A</math> and <math display=inline>\bigcap_{A\in \mathcal S} A</math> are commonly replaced with a notation involving indexed families, namely <math display=block>\bigcup_{i\in \mathcal I} A_i=\{x\mid (\exists i\in \mathcal I)\; x\in A_i\}</math> and <math display=block>\bigcap_{i\in \mathcal I} A_i=\{x\mid (\forall i\in \mathcal I)\; x\in A_i\}.</math>
The formulas of the above sections are special cases of the formulas for indexed families, where Template:Tmath and Template:Tmath. The formulas remain correct, even in the case where Template:Tmath for some Template:Tmath, since Template:Tmath
External operationsEdit
In Template:Alink, all elements of sets produced by set operations belong to previously defined sets. In this section, other set operations are considered, which produce sets whose elements can be outside all previously considered sets. These operations are Cartesian product, disjoint union, set exponentiation and power set.
Cartesian productEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Cartesian product of two sets has already be used for defining functions.
Given two sets Template:Tmath and Template:Tmath, their Cartesian product, denoted Template:Tmath is the set formed by all ordered pairs Template:Tmath such that Template:Tmath and Template:Tmath; that is, <math display=block>A_1\times A_2 = \{(a_1, a_2) \mid a_1\in A_1 \land a_2\in A_2\}.</math>
This definition does not supposes that the two sets are different. In particular, <math display=block>A\times A = \{(a_1, a_2) \mid a_1\in A \land a_2\in A\}.</math>
Since this definition involves a pair of indices (1,2), it generalizes straightforwardly to the Cartesian product or direct product of any indexed family of sets: <math display=block>\prod_{i\in \mathcal I} A_i= \{(a_i)_{i\in \mathcal I}\mid (\forall i\in \mathcal I) \;a_i\in A_i\}.</math> That is, the elements of the Cartesian product of a family of sets are all families of elements such that each one belongs to the set of the same index. The fact that, for every indexed family of nonempty sets, the Cartesian product is a nonempty set is insured by the axiom of choice.
Set exponentiationEdit
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Given two sets Template:Tmath and Template:Tmath, the set exponentiation, denoted Template:Tmath, is the set that has as elements all functions from Template:Tmath to Template:Tmath.
Equivalently, Template:Tmath can be viewed as the Cartesian product of a family, indexed by Template:Tmath, of sets that are all equal to Template:Tmath. This explains the terminology and the notation, since exponentiation with integer exponents is a product where all factors are equal to the base.
Power setEdit
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The power set of a set Template:Tmath is the set that has all subsets of Template:Tmath as elements, including the empty set and Template:Tmath itself.<ref name="Lucas1990" /> It is often denoted Template:Tmath. For example, <math display=block> \mathcal P(\{1,2,3\})=\{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}.</math>
There is a natural one-to-one correspondence (bijection) between the subsets of Template:Tmath and the functions from Template:Tmath to Template:Tmath; this correspondence associates to each subset the function that takes the value Template:Tmath on the subset and Template:Tmath elsewhere. Because of this correspondence, the power set of Template:Tmath is commonly identified with a set exponentiation: <math display=block> \mathcal P(E)=\{0,1\}^E.</math> In this notation, Template:Tmath is often abbreviated as Template:Tmath, which gives<ref name="Lucas1990" />Template:Sfn <math display=block> \mathcal P(E)=2^E.</math> In particular, if Template:Tmath has Template:Tmath elements, then Template:Tmath has Template:Tmath elements.Template:Sfn
Disjoint unionEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The disjoint union of two or more sets is similar to the union, but, if two sets have elements in common, these elements are considered as distinct in the disjoint union. This is obtained by labelling the elements by the indexes of the set they are coming from.
The disjoint union of two sets Template:Tmath and Template:Tmath is commonly denoted Template:Tmath and is thus defined as <math display=block>A\sqcup B=\{(a,i)\mid (i=1 \land a\in A)\lor (i=2 \land a\in B\}.</math>
If Template:Tmath is a set with Template:Tmath elements, then Template:Tmath has Template:Tmath elements, while Template:Tmath has Template:Tmath elements.
The disjoint union of two sets is a particular case of the disjoint union of an indexed family of sets, which is defined as <math display=block>\bigsqcup_{i \in \mathcal I}=\{(a,i)\mid i\in \mathcal I \land a\in A_i\}.</math>
The disjoint union is the coproduct in the category of sets. Therefore the notation <math display=block>\coprod_{i \in \mathcal I}=\{(a,i)\mid i\in \mathcal I \land a\in A_i\}</math> is commonly used.
Internal disjoint unionEdit
Given an indexed family of sets Template:Tmath, there is a natural map <math display=block>\begin{align} \bigsqcup_{i\in \mathcal I} A_i&\to \bigcup_{i\in \mathcal I} A_i\\ (a,i)&\mapsto a , \end{align}</math> which consists in "forgetting" the indices.
This maps is always surjective; it is bijective if and only if the Template:Tmath are pairwise disjoint, that is, all intersections of two sets of the family are empty. In this case, <math display=inline>\bigcup_{i\in \mathcal I} A_i</math> and <math display=inline>\bigsqcup_{i\in \mathcal I} A_i</math> are commonly identified, and one says that their union is the disjoint union of the members of the family.
If a set is the disjoint union of a family of subsets, one says also that the family is a partition of the set.
CardinalityEdit
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Informally, the cardinality of a set Template:Math, often denoted Template:Math, is the number of its members.<ref name="Moschovakis1994">Template:Cite book</ref> This number is the natural number Template:Tmath when there is a bijection between the set that is considered and the set Template:Tmath of the Template:Tmath first natural numbers. The cardinality of the empty set is Template:Tmath.<ref name="Smith2008">Template:Cite book</ref> A set with the cardinality of a natural number is called a finite set which is true for both cases. Otherwise, one has an infinite set.<ref>Template:Cite book</ref>
The fact that natural numbers measure the cardinality of finite sets is the basis of the concept of natural number, and predates for several thousands years the concept of sets. A large part of combinatorics is devoted to the computation or estimation of the cardinality of finite sets.
Infinite cardinalitiesEdit
The cardinality of an infinite set is commonly represented by a cardinal number, exactly as the number of elements of a finite set is represented by a natural numbers. The definition of cardinal numbers is too technical for this article; however, many properties of cardinalities can be dealt without referring to cardinal numbers, as follows.
Two sets Template:Tmath and Template:Tmath have the same cardinality if there exists a one-to-one correspondence (bijection) between them. This is denoted <math>|S|=|T|,</math> and would be an equivalence relation on sets, if a set of all sets would exist.
For example, the natural numbers and the even natural numbers have the same cardinality, since multiplication by two provides such a bijection. Similarly, the interval Template:Tmath and the set of all real numbers have the same cardinality, a bijection being provided by the function Template:Tmath.
Having the same cardinality of a proper subset is a characteristic property of infinite sets: a set is infinite if and only if it has the same cardinality as one of its proper subsets. So, by the above example, the natural numbers form an infinite set.<ref name="Lucas1990"/>
Besides equality, there is a natural inequality between cardinalities: a set Template:Tmath has a cardinality smaller than or equal to the cardinality of another set Template:Tmath if there is an injection frome Template:Tmath to Template:Tmath. This is denoted <math>|S|\le |T|.</math>
Schröder–Bernstein theorem implies that <math>|S|\le |T|</math> and <math>|T|\le |S|</math> imply <math>|S|= |T|.</math> Also, one has <math>|S|\le |T|,</math> if and only if there is a surjection from Template:Tmath to Template:Tmath. For every two sets Template:Tmath and Template:Tmath, one has either <math>|S|\le |T|</math> or <math>|T|\le |S|.</math>Template:Efn So, inequality of cardinalities is a total order.
The cardinality of the set Template:Tmath of the natural numbers, denoted <math>|\N|=\aleph_0,</math> is the smallest infinite cardinality. This means that if Template:Tmath is a set of natural numbers, then either Template:Tmath is finite or <math>|S|=|\N|.</math>
Sets with cardinality less than or equal to <math>|\N|=\aleph_0</math> are called countable sets; these are either finite sets or countably infinite sets (sets of cardinality <math>\aleph_0</math>); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than <math>\aleph_0</math> are called uncountable sets.
Cantor's diagonal argument shows that, for every set Template:Tmath, its power set (the set of its subsets) Template:Tmath has a greater cardinality: <math display=block>|S|<\left|2^S \right|.</math> This implies that there is no greatest cardinality.
Cardinality of the real numbersEdit
The cardinality of set of the real numbers is called the cardinality of the continuum and denoted Template:Tmath. (The term "continuum" referred to the real line before the 20th century, when the real line was not commonly viewed as a set of numbers.)
Since, as seen above, the real line Template:Tmath has the same cardinality of an open interval, every subset of Template:Tmath that contains a nonempty open interval has also the cardinality Template:Tmath.
One has <math display=block>\mathfrak c = 2^{\aleph_0},</math> meaning that the cardinality of the real numbers equals the cardinality of the power set of the natural numbers. In particular,<ref name="Stillwell2013">Template:Cite book</ref> <math display=block>\mathfrak c > \aleph_0.</math>
When published in 1878 by Georg Cantor,<ref name = "Cantor1878" /> this result was so astonishing that it was refused by mathematicians, and several tens years were needed before its common acceptance.
It can be shown that Template:Tmath is also the cardinality of the entire plane, and of any finite-dimensional Euclidean space.<ref name="Tall2006">Template:Cite book</ref>
The continuum hypothesis, was a conjecture formulated by Georg Cantor in 1878 that there is no set with cardinality strictly between Template:Tmath and Template:Tmath.<ref name = "Cantor1878">Template:Cite journal</ref> In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice.<ref name = "Cohen1963a"> Template:Cite journal </ref> This means that if the most widely used set theory is consistent (that is not self-contradictory),Template:Efn then the same is true for both the set theory with the continuum hypothesis added as a further axiom, and the set theory with the negation of the continuum hypothesis added.
Axiom of choiceEdit
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Informally, the axiom of choice says that, given any family of nonempty sets, one can choose simultaneously an element in each of them.Template:Efn Formulated this way, acceptability of this axiom sets a foundational logical question, because of the difficulty of conceiving an infinite instantaneous action. However, there are several equivalent formulations that are much less controversial and have strong consequences in many areas of mathematics. In the present days, the axiom of choice is thus commonly accepted in mainstream mathematics.
A more formal statement of the axiom of choice is: the Cartesian product of every indexed family of nonempty sets is non empty.
Other equivalent forms are described in the following subsections.
Zorn's lemmaEdit
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Zorn's lemma is an assertion that is equivalent to the axiom of choice under the other axioms of set theory, and is easier to use in usual mathematics.
Let Template:Tmath be a partial ordered set. A chain in Template:Tmath is a subset that is totally ordered under the induced order. Zorn's lemma states that, if every chain in Template:Tmath has an upper bound in Template:Tmath, then Template:Tmath has (at least) a maximal element, that is, an element that is not smaller than another element of Template:Tmath.
In most uses of Zorn's lemma, Template:Tmath is a set of sets, the order is set inclusion, and the upperbound of a chain is taken as the union of its members.
An example of use of Zorn's lemma, is the proof that every vector space has a basis. Here the elements of Template:Tmath are linearly independent subsets of the vector space. The union of a chain of elements of Template:Tmath is linearly independent, since an infinite set is linearly independent if and only if each finite subset is, and every finite subset of the union of a chain must be included in a member of the chain. So, there exist a maximal linearly independent set. This linearly independant set must span the vector space because of maximality, and is therefore a basis.
Another classical use of Zorn's lemma is the proof that every proper idealTemplate:Mdashthat is, an ideal that is not the whole ringTemplate:Mdashof a ring is contained in a maximal ideal. Here, Template:Tmath is the set of the proper ideals containing the given ideal. The union of chain of ideals is an ideal, since the axioms of an ideal involve a finite number of elements. The union of a chain of proper ideals is a proper ideal, since otherwise Template:Tmath would belong to the union, and this implies that it would belong to a member of the chain.
Transfinite inductionEdit
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The axiom of choice is equivalent with the fact that a well-order can be defined on every set, where a well-order is a total order such that every nonempty subset has a least element.
Simple examples of well-ordered sets are the natural numbers (with the natural order), and, for every Template:Mvar, the set of the Template:Mvar-tuples of natural numbers, with the lexicographic order.
Well-orders allow a generalization of mathematical induction, which is called transfinite induction. Given a property (predicate) Template:Tmath depending on a natural number, mathematical induction is the fact that for proving that Template:Tmath is always true, it suffice to prove that for every Template:Tmath,
- <math>(m<n \implies P(m)) \implies P(n).</math>
Transfinite induction is the same, replacing natural numbers by the elements of a well-ordered set.
Often, a proof by transfinite induction easier if three cases are proved separately, the two first cases being the same as for usual induction:
- <math>P(0)</math> is true, where Template:Tmath denotes the least element of the well-ordered set
- <math>P(x) \implies P(S(x)),\quad</math> where Template:Tmath denotes the successor of Template:Tmath, that is the least element that is greater than Template:Tmath
- <math>(\forall y;\; y<x \implies P(y)) \implies P(x) ,\quad</math> when Template:Tmath is not a successor.
Transfinite induction is fundamental for defining ordinal numbers and cardinal numbers.
See alsoEdit
- Algebra of sets
- Alternative set theory
- Category of sets
- Class (set theory)
- Family of sets
- Fuzzy set
- Mereology
- Principia Mathematica
NotesEdit
CitationsEdit
ReferencesEdit
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External linksEdit
- Template:Wiktionaryinline
- Cantor's "Beiträge zur Begründung der transfiniten Mengenlehre" Template:In lang
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