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| | 43 year-old Contract Administrator William from Elora, has several passions which include pets, tires for sale and collecting antiques. Felt particulary motivated after making a vacation in Port of the Moon.<br><br>Here is my weblog :: [http://ow.ly/AhDNp purchase tires] |
| [[File:Venn0111.svg|thumb|150px|Union of two sets:<br><math>~A \cup B</math>]]
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| [[File:Venn 0111 1111.svg|thumb|150px|Union of three sets:<br><math>~A \cup B \cup C</math>]]
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| In [[set theory]], the '''union''' (denoted by ∪) of a collection of sets is the set of all distinct [[element (set theory)|element]]s in the collection.<ref>{{cite web|url=http://mathworld.wolfram.com/Union.html|title=Union|author=Weisstein, Eric W|publisher=Wolfram's Mathworld|accessdate=2009-07-14}}</ref> It is one of the fundamental operations through which sets can be combined and related to each other.
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| == Union of two sets ==
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| The union of two sets ''A'' and ''B'' is the collection of points which are in ''A'' or in ''B'' or in both ''A'' and ''B''. In symbols,
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| :<math>A \cup B = \{ x: x \in A \,\,\,\textrm{ or }\,\,\, x \in B\}</math>.
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| For example, if ''A'' = {1, 3, 5, 7} and ''B'' = {1, 2, 4, 6} then ''A'' ∪ ''B'' = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is:
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| : ''A'' = {''x'' is an even [[integer]] larger than 1}
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| : ''B'' = {''x'' is an odd integer larger than 1}
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| : <math>A \cup B = \{2,3,4,5,6, \dots\}</math>
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| If we are then to refer to a single element by the variable "''x''", then we can say that ''x'' is a member of the union if it is an element present in set ''A'' or in set ''B'', or both.
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| Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the [[cardinality]] of a set or its contents. The number 9 is ''not'' contained in the union of the set of [[prime number]]s {2, 3, 5, 7, 11, …} and the set of [[even number]]s {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.
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| == Algebraic properties ==
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| Binary union is an [[associative]] operation; that is,
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| :''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.
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| The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e., either of the above can be expressed equivalently as ''A'' ∪ ''B'' ∪ ''C'').
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| Similarly, union is [[commutative]], so the sets can be written in any order.
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| The [[empty set]] is an [[identity element]] for the operation of union.
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| That is, ''A'' ∪ ∅ = ''A'', for any set ''A''.
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| These facts follow from analogous facts about [[logical disjunction]].
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| == Finite unions ==
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| One can take the union of several sets simultaneously. For example,
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| the union of three sets ''A'', ''B'', and ''C'' contains all elements of ''A'', all elements of ''B'', and all elements of ''C'', and nothing else.
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| Thus, ''x'' is an element of ''A'' ∪ ''B'' ∪ ''C'' if and only if ''x'' is in at least one of ''A'', ''B'', and ''C''.
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| In mathematics a '''finite union''' means any union carried out on a finite number of sets: it doesn't imply that the union set is a [[finite set]].
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| == Arbitrary unions ==
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| The most general notion is the union of an arbitrary collection of sets, sometimes called an ''infinitary union''.
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| If '''M''' is a set whose elements are themselves sets, then ''x'' is an element of the union of '''M''' [[if and only if]] there is [[existential quantification|at least one]] element ''A'' of '''M''' such that ''x'' is an element of ''A''.
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| In symbols:
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| : <math>x \in \bigcup\mathbf{M} \iff \exists A \in \mathbf{M},\ x \in A.</math>
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| That this union of '''M''' is a set no matter how large a set '''M''' itself might be, is the content of the [[axiom of union]] in [[axiomatic set theory]].
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| This idea subsumes the preceding sections, in that (for example) ''A'' ∪ ''B'' ∪ ''C'' is the union of the collection {''A'',''B'',''C''}.
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| Also, if '''M''' is the empty collection, then the union of '''M''' is the empty set.
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| The analogy between finite unions and logical disjunction extends to one between arbitrary unions and [[existential quantification]].
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| === Notations ===
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| The notation for the general concept can vary considerably. For a finite union of sets <math>S_1, S_2, S_3, \dots , S_n\,\!</math> one often writes <math>S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n</math>. Various common notations for arbitrary unions include <math>\bigcup \mathbf{M}</math>, <math>\bigcup_{A\in\mathbf{M}} A</math>, and <math>\bigcup_{i\in I} A_{i}</math>, the last of which refers to the union of the collection <math>\left\{A_i : i \in I\right\}</math> where ''I'' is an [[index set]] and <math>A_i</math> is a set for every <math>i \in I</math>.
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| In the case that the index set ''I'' is the set of [[natural number]]s, one uses a notation <math>\bigcup_{i=1}^{\infty} A_{i}</math> analogous to that of the [[series (mathematics)|infinite series]]. When formatting is difficult, this can also be written "''A''<sub>1</sub> ∪ ''A''<sub>2</sub> ∪ ''A''<sub>3</sub> ∪ ···".
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| (This last example, a union of countably many sets, is very common in [[analysis (math)|analysis]]; for an example see the article on [[sigma algebra|σ-algebras]].)
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| Whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.
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| === Union and intersection ===
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| [[Intersection (set theory)|Intersection]] distributes over union, in the sense that
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| : <math>A \cap \bigcup_{i\in I} B_{i} = \bigcup_{i\in I} (A \cap B_{i}).</math>
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| Within a given [[universe (mathematics)|universal set]], union can be written in terms of the operations of intersection and [[complement (set theory)|complement]] as
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| :<math>A \cup B = \left(A^C \cap B^C \right)^C</math> | |
| where the superscript <sup>C</sup> denotes the complement with respect to the universal set.
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| Arbitrary union and intersection also satisfy the law
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| : <math>\bigcup_{i\in I} \bigg(\bigcap_{j\in J} A_{i,j}\bigg) \subseteq \bigcap_{j\in J} \bigg(\bigcup_{i\in I} A_{i,j}\bigg)</math>. | |
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| == See also ==
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| * [[Alternation (formal language theory)]], the union of sets of strings
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| * [[Cardinality]]
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| * [[Complement (set theory)]]
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| * [[Disjoint union]]
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| * [[Intersection (set theory)]]
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| * [[Iterated binary operation]]
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| * [[Naive set theory]]
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| * [[Symmetric difference]]
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| ==Notes ==
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| <references/>
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| == External links ==
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| {{commons category}}
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| *{{MathWorld |title=Union |id=Union }}
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| *{{springer|title=Union of sets|id=p/u095390}}
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| *[http://www.apronus.com/provenmath/sum.htm Infinite Union and Intersection at ProvenMath] De Morgan's laws formally proven from the axioms of set theory.
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| {{Set theory}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Binary operations]]
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43 year-old Contract Administrator William from Elora, has several passions which include pets, tires for sale and collecting antiques. Felt particulary motivated after making a vacation in Port of the Moon.
Here is my weblog :: purchase tires