86.152.239.18: /* See also */ include Statistical model and use full name for Dynamics of Markovian Particles and reorder

2015-01-13T11:35:24Z

See also: include Statistical model and use full name for Dynamics of Markovian Particles and reorder

Show changes

en>Wavelength: removing 1 of "The"—MOS:HEAD

2014-02-18T03:11:06Z

removing 1 of "The"—MOS:HEAD

Show changes

en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

2014-01-30T10:42:40Z

Further reading: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 12:42, 30 January 2014
Line 1:		Line 1:
	~~Hello~~ and ~~welcome~~. ~~My name~~ is ~~Irwin~~ and ~~I totally dig~~ that ~~title~~. ~~To collect coins~~ is a ~~factor~~ that I'~~m totally addicted~~ to. ~~California is exactly~~ where I'~~ve always been residing~~ and I ~~love each working day living right here~~. ~~My working day job~~ is a ~~meter reader~~.<br><br>~~My blog~~: ~~healthy food delivery~~ ([http://~~Mwp~~.mx/~~dietmeals23439 please click~~ the ~~next website page~~])		{{insufficient inline citations\|date=July 2013}}
			[[File:Venn0111.svg\|thumb\|150px\|Union of two sets:<br><math>~A \cup B</math>]]
			[[File:Venn 0111 1111.svg\|thumb\|150px\|Union of three sets:<br><math>~A \cup B \cup C</math>]]
			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.

			== Union of two sets ==
			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,

			:<math>A \cup B = \{ x: x \in A \,\,\,\textrm{ or }\,\,\, x \in B\}</math>.

			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:
			: ''A'' = {''x'' is an even [[integer]] larger than 1}
			: ''B'' = {''x'' is an odd integer larger than 1}
			: <math>A \cup B = \{2,3,4,5,6, \dots\}</math>

			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.

			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.

			== Algebraic properties ==
			Binary union is an [[associative]] operation; that is,

			:''A'' ∪ (''B'' ∪ ''C'') = (''A'' ∪ ''B'') ∪ ''C''.

			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'').
			Similarly, union is [[commutative]], so the sets can be written in any order.

			The [[empty set]] is an [[identity element]] for the operation of union.
			That is, ''A'' ∪ ∅ = ''A'', for any set ''A''.

			These facts follow from analogous facts about [[logical disjunction]].

			== Finite unions ==
			One can take the union of several sets simultaneously. For example,
			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.
			Thus, ''x'' is an element of ''A'' ∪ ''B'' ∪ ''C'' if and only if ''x'' is in at least one of ''A'', ''B'', and ''C''.

			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]].

			== Arbitrary unions ==

			The most general notion is the union of an arbitrary collection of sets, sometimes called an ''infinitary union''.
			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''.
			In symbols:
			: <math>x \in \bigcup\mathbf{M} \iff \exists A \in \mathbf{M},\ x \in A.</math>
			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]].

			This idea subsumes the preceding sections, in that (for example) ''A'' ∪ ''B'' ∪ ''C'' is the union of the collection {''A'',''B'',''C''}.
			Also, if '''M''' is the empty collection, then the union of '''M''' is the empty set.
			The analogy between finite unions and logical disjunction extends to one between arbitrary unions and [[existential quantification]].

			=== Notations ===
			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>.
			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> ∪ ···".
			(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]].)

			Whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.

			=== Union and intersection ===
			[[Intersection (set theory)\|Intersection]] distributes over union, in the sense that
			: <math>A \cap \bigcup_{i\in I} B_{i} = \bigcup_{i\in I} (A \cap B_{i}).</math>

			Within a given [[universe (mathematics)\|universal set]], union can be written in terms of the operations of intersection and [[complement (set theory)\|complement]] as
			:<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.

			Arbitrary union and intersection also satisfy the law
			: <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>.

			== See also ==
			* [[Alternation (formal language theory)]], the union of sets of strings
			* [[Cardinality]]
			* [[Complement (set theory)]]
			* [[Disjoint union]]
			* [[Intersection (set theory)]]
			* [[Iterated binary operation]]
			* [[Naive set theory]]
			* [[Symmetric difference]]

			==Notes ==
			<references/>

			== External links ==
			{{commons category}}
			*{{MathWorld \|title=Union \|id=Union }}
			*{{springer\|title=Union of sets\|id=p/u095390}}
			*[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.

			{{Set theory}}

			[[Category:Basic concepts in set theory]]
			[[Category:Binary operations]]

en>JackieBot: r2.7.2) (Robot: Adding et:Juhuslik protsess

2012-07-21T15:55:43Z

r2.7.2) (Robot: Adding et:Juhuslik protsess

New page

Hello and welcome. My name is Irwin and I totally dig that title. To collect coins is a factor that I'm totally addicted to. California is exactly where I've always been residing and I love each working day living right here. My working day job is a meter reader.<br><br>My blog: healthy food delivery ([http://Mwp.mx/dietmeals23439 please click the next website page])

Stochastic process - Revision history

86.152.239.18: /* See also */ include Statistical model and use full name for Dynamics of Markovian Particles and reorder

en>Wavelength: removing 1 of "The"—MOS:HEAD

en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

en>JackieBot: r2.7.2) (Robot: Adding et:Juhuslik protsess