|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{about|what mathematicians call "intuitive" or "naive" set theory|a more detailed account|Naive set theory|a rigorous modern [[axiom]]atic treatment of sets|Set theory}}
| | == do not sound a hundred heart back into his head == |
|
| |
|
| [[File:Venn A intersect B.svg|right|thumb|An example of a [[Venn diagram]]<br />The ''intersection'' of two sets is made up with the objects contained in both sets]]
| | Joseph was a smile, stunned to catch up to Lee Mei asked: '? How can you go.'<br><br>guarantee is I do sin hands, here I can simply reach of sin [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペードニューヨーク 財布] ah, Caoya Jie hit a key, with one mind and said: 'Do not forget brother is clairvoyant company CEOs ......... their monitoring equipment is to go [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード 財布 セール] through [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ] our public security and acceptance, and even many insiders that we sell to their equipment, you say such a [http://www.dmwai.com/webalizer/kate-spade-14.html ケイトスペード ハンドバッグ] device, how can the experts stumped me, want to know how the back door into it? '<br><br>He proudly said, Hey, do not sound a hundred heart back into his head, only to find all gathered behind him, and Yu Feng chuckled and asked: '? into the back door of how it feels.'<br><br>'yo, Caoge have this hobby.' mouse Road with, Li Mei, biting his lower lip angry, afraid to talk to each group of hooligans.<br><br>Caoya Jie cheap cheap smile, a touch of handsome hair, black jack knock an Enter key, a remote execution of the program, brush brush lit screen, a synchronized to [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード 財布 ゴールド] a monitoring unit here, mouse stunned with: 'Oh, This feeling is the back door into the great. |
| | | 相关的主题文章： |
| In [[mathematics]], a '''set''' is a collection of distinct objects, considered as an [[mathematical object|object]] in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In [[mathematics education]], elementary topics such as [[Venn diagram]]s are taught at a young age, while more advanced concepts are taught as part of a university degree. The term itself was coined by Bolzano in his work [[The Paradoxes of the Infinite]].
| | <ul> |
| | | |
| ==Definition==
| | <li>[http://ka.17p.com/plus/feedback.php?aid=2651 http://ka.17p.com/plus/feedback.php?aid=2651]</li> |
| A set is a well defined collection of distinct objects. The objects that make up a set (also known as the [[element (mathematics)|element]]s or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. [[Georg Cantor]], the founder of set theory, gave the following definition of a set at the beginning of his ''Beiträge zur Begründung der transfiniten Mengenlehre'':<ref>"Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener Objekte unserer Anschauung oder unseres Denkens – welche Elemente der Menge genannt werden – zu einem Ganzen." [http://www.brinkmann-du.de/mathe/fos/fos01_03.htm]</ref>
| | |
| | | <li>[http://kuppingercabin.com/cgi-bin/kupcabin/mk_cal.cgi http://kuppingercabin.com/cgi-bin/kupcabin/mk_cal.cgi]</li> |
| {{quote|A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought – which are called [[element (mathematics)|elements]] of the set.}}
| | |
| | | <li>[http://www.kangmeizyr.com/forum.php?mod=viewthread&tid=71541&fromuid=28107 http://www.kangmeizyr.com/forum.php?mod=viewthread&tid=71541&fromuid=28107]</li> |
| Sets are conventionally denoted with [[capital letters]]. Sets ''A'' and ''B'' are equal [[if and only if]] they have precisely the same elements.<ref name="Stoll">
| | |
| {{Cite book | last = Stoll | first = Robert | authorlink = | coauthors = | title = Sets, Logic and Axiomatic Theories | publisher = W. H. Freeman and Company | series = | volume = | edition = | date = | location = | pages = 5 | language = | url = | doi = | id = | isbn = | mr = | zbl = | jfm = }}</ref>
| | </ul> |
| | |
| As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an [[undefined primitive]] in [[axiomatic set theory]], and its properties are defined by the [[Zermelo–Fraenkel axioms]]. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if every element of one is an element of the other.
| |
| | |
| ==Describing sets==
| |
| There are two ways of describing, or specifying the members of, a set. One way is by [[intensional definition]], using a rule or semantic description:
| |
| | |
| :''A'' is the set whose members are the first four positive [[integer]]s. | |
| :''B'' is the set of colors of the [[Flag of France|French flag]].
| |
| | |
| The second way is by [[extension (semantics)|extension]] – that is, listing each member of the set. An [[extensional definition]] is denoted by enclosing the list of members in [[Bracket#Curly_brackets_.7B_.7D|curly brackets]]:
| |
| | |
| :''C'' = {4, 2, 1, 3}
| |
| :''D'' = {blue, white, red}.
| |
| | |
| Every element of a set must be unique; no two members may be identical. (A [[multiset]] is a generalized concept of a set that relaxes this criterion.) All [[set operations (mathematics)|set operations]] preserve this property. The order in which the elements of a set or multiset are listed is irrelevant (unlike for a [[sequence]] or [[tuple]]). Combining these two ideas into an example
| |
| :{6, 11} = {11, 6} = {11, 6, 6, 11} | |
| because the extensional specification means merely that each of the elements listed is a member of the set.
| |
| | |
| For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as:
| |
| | |
| :{1, 2, 3, ..., 1000},
| |
| | |
| where the [[ellipsis]] ("...") indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive [[even number]]s can be written as {{nowrap|{2, 4, 6, 8, ... }.}}
| |
| | |
| The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have the meaning "the set of all ...". So, ''E'' = {playing card suits} is the set whose four members are {{nowrap|♠, ♦, ♥, and ♣.}} A more general form of this is [[set-builder notation]], through which, for instance, the set ''F'' of the twenty smallest integers that are four less than [[square number|perfect square]]s can be denoted:
| |
| | |
| :''F'' = {''n''<sup>2</sup> − 4 : ''n'' is an integer; and 0 ≤ ''n'' ≤ 19}.
| |
| | |
| In this notation, the [[Colon (punctuation)|colon]] (":") means "such that", and the description can be interpreted as "''F'' is the set of all numbers of the form ''n''<sup>2</sup> − 4, such that ''n'' is a whole number in the range from 0 to 19 inclusive." Sometimes the [[vertical bar]] ("|") is used instead of the colon.
| |
| | |
| One often has the choice of specifying a set intensionally or extensionally. In the examples above, for instance, ''A'' = ''C'' and ''B'' = ''D''.
| |
| | |
| ==Membership==
| |
| {{Main|Element (mathematics)}}
| |
| The key relation between sets is ''membership'' – when one set is an element of another. If ''a'' is a member of ''B'', this is denoted ''a'' ∈ ''B'', while if ''c'' is not a member of ''B'' then ''c'' ∉ ''B''.
| |
| For example, with respect to the sets ''A'' = {1,2,3,4}, ''B'' = {blue, white, red}, and ''F'' = {''n''<sup>2</sup> − 4 : ''n'' is an integer; and 0 ≤ ''n'' ≤ 19} defined above,
| |
| :4 ∈ ''A'' and 12 ∈ ''F''; but
| |
| :9 ∉ ''F'' and green ∉ ''B''.
| |
| | |
| ===Subsets===
| |
| {{Main|Subset}}
| |
| If every member of set ''A'' is also a member of set ''B'', then ''A'' is said to be a ''subset'' of ''B'', written ''A'' ⊆ ''B'' (also pronounced ''A is contained in B''). Equivalently, we can write ''B'' ⊇ ''A'', read as ''B is a superset of A'', ''B includes A'', or ''B contains A''. The [[relation (mathematics)|relationship]] between sets established by ⊆ is called ''inclusion'' or ''containment''.
| |
| | |
| If ''A'' is a subset of, but not equal to, ''B'', then ''A'' is called a ''proper subset'' of ''B'', written ''A'' ⊊ ''B'' (''A is a proper subset of B'') or ''B'' ⊋ ''A'' (''B is a proper superset of A'').
| |
| | |
| Note that the expressions ''A'' ⊂ ''B'' and ''B'' ⊃ ''A'' are used differently by different authors; some authors use them to mean the same as ''A'' ⊆ ''B'' (respectively ''B'' ⊇ ''A''), whereas other use them to mean the same as ''A'' ⊊ ''B'' (respectively ''B'' ⊋ ''A'').
| |
| | |
| <div style="float:right;margin:1em;">[[File:Venn A subset B.svg|150px|center|A is a subset of B]]<div class="center"><small> ''A'' is a '''subset''' of ''B''</small></div></div>
| |
| | |
| Example:
| |
| :* The set of all men is a proper [[subset]] of the set of all people.
| |
| :* {1, 3} ⊊ {1, 2, 3, 4}.
| |
| :* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
| |
| | |
| The empty set is a subset of every set and every set is a subset of itself:
| |
| :* ∅ ⊆ ''A''.
| |
| :* ''A'' ⊆ ''A''.
| |
| | |
| An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:
| |
| :* {{nowrap|1=''A'' = ''B''}} if and only if {{nowrap|''A'' ⊆ ''B''}} and {{nowrap|''B'' ⊆ ''A''}}.
| |
| | |
| A [[partition of a set]] ''S'' is a set of nonempty subsets of ''S'' such that every element ''x'' in ''S'' is in exactly one of these subsets.
| |
| | |
| ===Power sets===
| |
| {{Main|Power set}}
| |
| The power set of a set ''S'' is the set of all subsets of ''S'', including ''S'' itself and the empty set. For example, the power set of the set {1, 2, 3} is <nowiki>{{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}</nowiki>. The power set of a set ''S'' is usually written as ''P''(''S'').
| |
| | |
| The power set of a finite set with ''n'' elements has 2<sup>''n''</sup> elements. This relationship is one of the reasons for the terminology ''power set''. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 2<sup>3</sup> = 8 elements.
| |
| | |
| The power set of an infinite (either [[countable]] or [[uncountable]]) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair the elements of a set ''S'' with the elements of its power set ''P''(''S'') such that every element of ''S'' set is paired with exactly one element of ''P''(''S''), and every element of ''P''(''S'') is paired with exactly one element of ''S''. (There is never a [[bijection]] from ''S'' onto ''P''(''S'').)
| |
| | |
| Every partition of a set ''S'' is a subset of the powerset of ''S''.
| |
| | |
| ==Cardinality==
| |
| {{Main|Cardinality}}
| |
| | |
| The cardinality | ''S'' | of a set ''S'' is "the number of members of ''S''." For example, if {{nowrap|1=''B'' = <nowiki>{</nowiki>''blue, white, red''<nowiki>}</nowiki>}}, {{nowrap|1={{!}} ''B'' {{!}} = 3.}}
| |
| | |
| There is a unique set with no members and zero cardinality, which is called the ''[[empty set]]'' (or the ''null set'') and is denoted by the symbol ∅ (other notations are used; see [[empty set]]). For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the [[0 (number)|number zero]], is important in mathematics; indeed, the existence of this set is one of the fundamental concepts of [[axiomatic set theory]].
| |
| | |
| Some sets have [[Infinite set|infinite]] cardinality. The set '''N''' of [[natural number]]s, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of [[real number]]s has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a [[straight line]] is the same as the cardinality of any [[line segment|segment]] of that line, of the entire [[plane (mathematics)|plane]], and indeed of any [[dimension (mathematics)|finite-dimensional]] [[Euclidean space]].
| |
| | |
| ==Special sets==
| |
| There are some sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the [[empty set]], denoted {} or ∅. Another is the [[unit set]] {x}, which contains exactly one element, namely x.<ref name="Stoll"/> Many of these sets are represented using [[blackboard bold]] or bold typeface. Special sets of numbers include:
| |
| * '''P''' or ℙ, denoting the set of all [[prime number|primes]]: '''P''' = {2, 3, 5, 7, 11, 13, 17, ...}.
| |
| * '''N''' or ℕ, denoting the set of all [[natural number]]s: '''N''' = {1, 2, 3, . . .} (sometimes defined containing 0).
| |
| * '''Z''' or ℤ, denoting the set of all [[integer]]s (whether positive, negative or zero): '''Z''' = {..., −2, −1, 0, 1, 2, ...}.
| |
| * '''Q''' or ℚ, denoting the set of all [[rational number]]s (that is, the set of all [[proper fraction|proper]] and [[improper fraction]]s): '''Q''' = {''a''/''b'' : ''a'', ''b'' ∈ '''Z''', ''b'' ≠ 0}. For example, 1/4 ∈ '''Q''' and 11/6 ∈ '''Q'''. All integers are in this set since every integer ''a'' can be expressed as the fraction ''a''/1 ('''Z''' ⊊ '''Q''').
| |
| * '''R''' or ℝ, denoting the set of all [[real number]]s. This set includes all rational numbers, together with all [[irrational number|irrational]] numbers (that is, numbers that cannot be rewritten as fractions, such as √<span style="text-decoration:overline">2</span>, as well as [[transcendental numbers]] such as [[Pi|π]], [[e (mathematical constant)|''e'']] and [[definable real number|numbers that cannot be defined]]).
| |
| * '''C''' or ℂ, denoting the set of all [[complex number]]s: '''C''' = {''a'' + ''bi'' : ''a'', ''b'' ∈ '''R'''}. For example, 1 + 2''i'' ∈ '''C'''.
| |
| * '''H''' or ℍ, denoting the set of all [[quaternion]]s: '''H''' = {''a'' + ''bi'' + ''cj'' + ''dk'' : ''a'', ''b'', ''c'', ''d'' ∈ '''R'''}. For example, 1 + ''i'' + 2''j'' − ''k'' ∈ '''H'''.
| |
| | |
| Positive and negative sets are denoted by a superscript - or +, for example: ℚ<sup>+</sup> represents the set of positive rational numbers.
| |
| | |
| Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of [[number theory]] and related fields.
| |
| | |
| ==Basic operations==
| |
| There are several fundamental operations for constructing new sets from given sets.
| |
| | |
| ===Unions===
| |
| [[File:Venn0111.svg|thumb|<div class="center">The '''union''' of ''A'' and ''B'', denoted {{nowrap|''A'' ∪ ''B''}}</div>]]
| |
| {{Main|Union (set theory)}}
| |
| Two sets can be "added" together. The ''union'' of ''A'' and ''B'', denoted by ''A'' ∪ ''B'', is the set of all things that are members of either ''A'' or ''B''.
| |
| | |
| Examples:
| |
| :* {{nowrap|1={1, 2} ∪ {1, 2} = {1, 2}. }}
| |
| :* {{nowrap|1={1, 2} ∪ {2, 3} = {1, 2, 3}. }}
| |
| | |
| Some basic properties of unions:
| |
| :* {{nowrap|1=''A'' ∪ ''B'' = ''B'' ∪ ''A''.}}
| |
| :* {{nowrap|1=''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.}}
| |
| :* {{nowrap|1=''A'' ⊆ (''A'' ∪ ''B'').}}
| |
| :* {{nowrap|1=''A'' ∪ ''A'' = ''A''.}}
| |
| :* {{nowrap|1=''A'' ∪ ∅ = ''A''.}}
| |
| :* {{nowrap|''A'' ⊆ ''B''}} [[if and only if]] {{nowrap|1=''A'' ∪ ''B'' = ''B''.}}
| |
| | |
| ===Intersections===
| |
| {{Main|Intersection (set theory)}}
| |
| A new set can also be constructed by determining which members two sets have "in common". The ''intersection'' of ''A'' and ''B'', denoted by {{nowrap|''A'' ∩ ''B'',}} is the set of all things that are members of both ''A'' and ''B''. If {{nowrap|1=''A'' ∩ ''B'' = ∅,}} then ''A'' and ''B'' are said to be ''disjoint''.
| |
| [[File:Venn0001.svg|thumb|<div class="center">The '''intersection''' of ''A'' and ''B'', denoted {{nowrap|''A'' ∩ ''B''.}}</div>]]
| |
| | |
| Examples:
| |
| :* {{nowrap|1={1, 2} ∩ {1, 2} = {1, 2}.}}
| |
| :* {{nowrap|1={1, 2} ∩ {2, 3} = {2}.}}
| |
| | |
| Some basic properties of intersections:
| |
| :* {{nowrap|1=''A'' ∩ ''B'' = ''B'' ∩ ''A''.}}
| |
| :* {{nowrap|1=''A'' ∩ (''B'' ∩ ''C'') = (''A'' ∩ ''B'') ∩ ''C''.}}
| |
| :* {{nowrap|''A'' ∩ ''B'' ⊆ ''A''.}}
| |
| :* {{nowrap|1=''A'' ∩ ''A'' = ''A''.}}
| |
| :* {{nowrap|1=''A'' ∩ ∅ = ∅.}}
| |
| :* {{nowrap|''A'' ⊆ ''B''}} [[if and only if]] {{nowrap| 1=''A'' ∩ ''B'' = ''A''.}}
| |
| | |
| ===Complements===
| |
| [[File:Venn0100.svg|thumb|<div class="center">The '''relative complement'''<br/>of ''B'' in ''A''</div>]]
| |
| [[File:Venn1010.svg|thumb|<div class="center">The '''complement''' of ''A'' in ''U''</div>]] | |
| [[File:Venn0110.svg|thumb|<div class="center">The '''symmetric difference''' of ''A'' and ''B''</div>]]
| |
| {{Main|Complement (set theory)}}
| |
| Two sets can also be "subtracted". The ''relative complement'' of ''B'' in ''A'' (also called the ''set-theoretic difference'' of ''A'' and ''B''), denoted by {{nowrap|''A'' \ ''B''}} (or {{nowrap|''A'' − ''B''}}), is the set of all elements that are members of ''A'' but not members of ''B''. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element ''green'' from the set {{nowrap|{1, 2, 3};}} doing so has no effect.
| |
| | |
| In certain settings all sets under discussion are considered to be subsets of a given [[universe (mathematics)|universal set]] ''U''. In such cases, {{nowrap|''U'' \ ''A''}} is called the ''absolute complement'' or simply ''complement'' of ''A'', and is denoted by ''A''′.
| |
| | |
| Examples:
| |
| :* {{nowrap|1={1, 2} \ {1, 2} = ∅.}}
| |
| :* {{nowrap|1={1, 2, 3, 4} \ {1, 3} = {2, 4}.}}
| |
| :* If ''U'' is the set of integers, ''E'' is the set of even integers, and ''O'' is the set of odd integers, then ''U'' \ ''E'' = ''E''′ = ''O''.
| |
| | |
| Some basic properties of complements:
| |
| :* {{nowrap|1=''A'' \ ''B'' ≠ ''B'' \ ''A''}} for {{nowrap|1=''A'' ≠ ''B''}}.
| |
| :* {{nowrap|1=''A'' ∪ ''A''′ = ''U''.}}
| |
| :* {{nowrap|1=''A'' ∩ ''A''′ = ∅.}}
| |
| :* {{nowrap|1=(''A''′)′ = ''A''.}}
| |
| :* {{nowrap|1=''A'' \ ''A'' = ∅.}}
| |
| :* {{nowrap|1=''U''′ = ∅}} and {{nowrap|1=∅′ = ''U''.}}
| |
| :* {{nowrap|1=''A'' \ ''B'' = ''A'' ∩ ''B''′}}.
| |
| | |
| An extension of the complement is the [[symmetric difference]], defined for sets ''A'', ''B'' as
| |
| :<math>A\,\Delta\,B = (A \setminus B) \cup (B \setminus A).</math>
| |
| For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}.
| |
| | |
| ===Cartesian product===
| |
| {{Main|Cartesian product}}
| |
| A new set can be constructed by associating every element of one set with every element of another set. The ''Cartesian product'' of two sets ''A'' and ''B'', denoted by ''A'' × ''B'' is the set of all [[ordered pairs]] (''a'', ''b'') such that ''a'' is a member of ''A'' and ''b'' is a member of ''B''.
| |
| | |
| Examples:
| |
| :* {{nowrap|1={1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.}}
| |
| :* {{nowrap|1={1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green) }.}} | |
| :* {{nowrap|1={1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.}}
| |
| | |
| Some basic properties of cartesian products:
| |
| :* {{nowrap|1=''A'' × [[∅]] = ∅.}}
| |
| :* {{nowrap|1=''A'' × (''B'' ∪ ''C'') = (''A'' × ''B'') ∪ (''A'' × ''C'').}}
| |
| :* {{nowrap|1=(''A'' ∪ ''B'') × ''C'' = (''A'' × ''C'') ∪ (''B'' × ''C'').}}
| |
| Let ''A'' and ''B'' be finite sets. Then
| |
| :* | ''A'' × ''B'' | = | ''B'' × ''A'' | = | ''A'' | × | ''B'' |.
| |
| <!-- Expand, please. -->
| |
| | |
| ==Applications==
| |
| Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, [[Algebraic structure|structures]] in [[abstract algebra]], such as [[Group (mathematics)|groups]], [[Field (mathematics)|fields]] and [[Ring (mathematics)|rings]], are sets [[Closure (mathematics)|closed]] under one or more operations.
| |
| | |
| One of the main applications of naive set theory is constructing [[Relation (mathematics)|relations]]. A relation from a [[domain (mathematics)|domain]] ''A'' to a [[codomain]] ''B'' is a subset of the Cartesian product ''A'' × ''B''. Given this concept, we are quick to see that the set ''F'' of all ordered pairs (''x'', ''x''<sup>2</sup>), where ''x'' is real, is quite familiar. It has a domain set '''R''' and a codomain set that is also '''R''', because the set of all squares is subset of the set of all reals. If placed in functional notation, this relation becomes ''f''(''x'') = ''x''<sup>2</sup>. The reason these two are equivalent is for any given value, ''y'' that the function is defined for, its corresponding ordered pair, (''y'', ''y''<sup>2</sup>) is a member of the set ''F''.
| |
| <!-- Expand -->
| |
| | |
| ==Axiomatic set theory==
| |
| {{Main|Axiomatic set theory}}
| |
| Although initially [[naive set theory]], which defines a set merely as ''any [[well-defined]]'' collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned [[:Category:Paradoxes of naive set theory|several paradoxes]], most notably:
| |
| *[[Russell's paradox]]—It shows that the "set of all sets that ''do not contain themselves''," i.e. the "set" { ''x'' : ''x'' is a set and ''x'' ∉ ''x'' } does not exist.
| |
| *[[Cantor's paradox]]—It shows that "the set of all sets" cannot exist.
| |
| | |
| The reason is that the phrase ''well-defined'' is not very well defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on [[first-order logic]], and thus '''[[axiomatic set theory]]''' was born.
| |
| | |
| For most purposes however, [[naive set theory]] is still useful.
| |
| | |
| ==Principle of inclusion and exclusion==
| |
| {{Main|Inclusion-exclusion principle}}
| |
| | |
| This principle gives us the cardinality of the union of sets.
| |
| <math>\begin{align}
| |
| \left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right|= & \left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots\left|A_{n}\right|\right)- \\
| |
| & \left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots\left|A_{n-1}\cap A_{n}\right|\right)+ \\
| |
| &\ldots+ \\
| |
| &\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right|\right)
| |
| \end{align}</math>
| |
| | |
| ==See also==
| |
| {{Portal|Logic}}
| |
| <div style="-moz-column-count:4; column-count:4;"> | |
| * [[Alternative set theory]]
| |
| * [[Axiomatic set theory]]
| |
| * [[Boolean algebra (logic)]]
| |
| * [[Category of sets]]
| |
| * [[Class (set theory)]]
| |
| * [[Dense set]]
| |
| * [[Family of sets]]
| |
| * [[Fuzzy set]]
| |
| * [[Internal set]]
| |
| * [[Mathematical structure]]
| |
| * [[Mereology]]
| |
| * [[Multiset]]
| |
| * [[Naive set theory]]
| |
| * [[Principia Mathematica]]
| |
| * [[Rough set]]
| |
| * [[Russell's paradox]]
| |
| * [[Scientific classification]]
| |
| * [[Sequence (mathematics)]]
| |
| * [[Set notation]]
| |
| * [[Taxonomy (general)|Taxonomy]]
| |
| * [[Tuple]]
| |
| </div>
| |
| | |
| ==Notes==
| |
| <!--<nowiki>
| |
| See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref> and </ref> tags, and the template below.
| |
| </nowiki>-->
| |
| {{Reflist}}
| |
| | |
| ==References== | |
| {{Commons|Sets}}
| |
| *[[Joseph Dauben|Dauben, Joseph W.]], ''Georg Cantor: His Mathematics and Philosophy of the Infinite'', Boston: [[Harvard University Press]] (1979) ISBN 978-0-691-02447-9.
| |
| *[[Paul Halmos|Halmos, Paul R.]], ''Naive Set Theory'', Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6.
| |
| *Stoll, Robert R., ''Set Theory and Logic'', Mineola, N.Y.: [[Dover Publications]] (1979) ISBN 0-486-63829-4.
| |
| *Velleman, Daniel, ''How To Prove It: A Structured Approach'', [[Cambridge University Press]] (2006) ISBN 978-0-521-67599-4
| |
| | |
| ==External links==
| |
| * [http://www.c2.com/cgi/wiki?SetTheory C2 Wiki – Examples of set operations using English operators.]
| |
| * [http://education-portal.com/academy/lesson/mathematical-sets-elements-intersections-unions.html Mathematical Sets: Elements, Intersections & Unions, Education Portal Academy]
| |
| {{Logic}}
| |
| {{Set theory}}
| |
| | |
| [[Category:Concepts in logic]]
| |
| [[Category:Mathematical objects]]
| |
| [[Category:Set theory|*]]
| |