
	Where results make sense

Topic: Axiom of union

Related Topics

axiom of choice

Separation axioms

Peano axioms

probability axioms

axiom of extensionality

Zermelo Fraenkel axioms

Axiom of regularity

Kuratowski closure axioms

Axiom schema of specification

axiom of infinity

axiom of empty set

Axiom of pairs

In the News (Sat 5 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Axiom of union - Wikipedia, the free encyclopedia
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x.
		The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		A as {C in B : for all D in A, C is in D} using the axiom schema of specification.
	en.wikipedia.org /wiki/Axiom_of_union (310 words)

	Union (set theory) - Wikipedia, the free encyclopedia
		The number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.
		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.
		The analogy between finite unions and logical disjunction extends to one between infinite unions and existential quantification.
	en.wikipedia.org /wiki/Union_(set_theory) (668 words)

	Axiomatic set theory - Wikipedia, the free encyclopedia
		Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
		Axiom of separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds.
		Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.
	en.wikipedia.org /wiki/Axiomatic_set_theory (2518 words)

	PlanetMath: axiom of union
		The Axiom of Union is an axiom of Zermelo-Fraenkel set theory.
		More succinctly, the union of any set of sets is a set.
		This is version 5 of axiom of union, born on 2003-06-25, modified 2003-06-26.
	planetmath.org /encyclopedia/Union2.html (94 words)

	Zermelo-Fraenkel set theory - the free encyclopedia (Site not responding. Last check: 2007-10-14)
		Axiom of pairing: If x, y are sets, then there exists a set containing x and y as its only elements, which we denote by {x,y} or {x} ∪ {y}.
		Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
		Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is y ∪ {y}.
	www.world-knowledge-encyclopedia.com /?t=ZFC (444 words)

	Knowledge King - Axiom of pairing (Site not responding. Last check: 2007-10-14)
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory.
		The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		Note that adopting this as an axiom schema will not replace the axiom of union, which is still needed for other situations.
	www.knowledgeking.net /encyclopedia/a/ax/axiom_of_pairing.html (477 words)

	Zermelo-Fraenkel set theory (Site not responding. Last check: 2007-10-14)
		The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory.
		The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Abraham Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 (Zermelo set theory).
		The axiom system is written in first-order logic; it has an infinite number of axioms because an axiom schema is used.
	www.1bx.com /en/ZFC.htm (513 words)

	Axiom schema of replacement -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-14)
		Suppose P is any ((logic) what is predicated of the subject of a proposition; the second term in a proposition is predicated of the first term by means of the copula) predicate in two (A quantity that can assume any of a set of values) variables that doesn't use the symbol B.
		Thus, what the axiom schema is really saying is that, given a set A, we can find a set B whose members are precisely the values of F at the members of A.
		But in this case, the set B required for the axiom of specification is the (Click link for more info and facts about empty set) empty set, so the axiom schema follows in general using also the (Click link for more info and facts about axiom of empty set) axiom of empty set.
	www.absoluteastronomy.com /encyclopedia/A/Ax/Axiom_schema_of_replacement.htm (1214 words)

	Axiom of union -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-14)
		By the (Click link for more info and facts about axiom of extensionality) axiom of extensionality this set B is unique and it is called the (The state of being joined or united or linked) union of A, and denoted
		The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative (Click link for more info and facts about axiomatization) axiomatization of set theory.
		Note that there is no corresponding axiom of (A junction where one street or road crosses another) intersection.
	www.absoluteastronomy.com /encyclopedia/A/Ax/Axiom_of_union.htm (258 words)

	Set theory - Wikipedia
		The appearance around the turn of the century of the so-called set-theoretical paradoxes, such as Russell's Paradox, prompted the formulation in 1908 by Ernst Zermelo of an axiomatic theory of sets.
		The axioms for set theory now most often studied and used are those called the Zermelo-Fraenkel axioms, usually together with the axiom of choice.
		The Zermelo-Fraenkel axioms are commonly abbreviated to ZF, or ZFC if the axiom of choice is included.
	nostalgia.wikipedia.org /wiki/SetTheory (674 words)

	Encyclopedia: Set theoretic union (Site not responding. Last check: 2007-10-14)
		The union of A and B is usually written "A ∪B".
		The number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.
		Similarly, union is commutative, so you can write the sets in any order.
	www.nationmaster.com /encyclopedia/Set-theoretic-union (684 words)

	Zermelo-Fraenkel set theory : ZFC
		The Zermelo-Fraenkel axioms of set theory, denoted ZF, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based.
		The axioms are the result of work by Thoralf Skolem[?] in 1922, based on earlier work by Adolf Fraenkel[?] in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 (Zermelo set theory[?]).
		The axiom system has an infinite number of axioms because an axiom schema[?] is used.
	www.fastload.org /zf/ZFC.html (442 words)

	axiom
		The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.
		As the word axiom is understood in mathematics, an axiom is not a proposition that is self-evident.
		An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.
	www.fact-library.com /axiom.html (667 words)

	Naive set theory - Wikipedia, the free encyclopedia
		Links in this article to specific axioms of set theory point out some of the relationships between the informal discussion here and the formal axiomatization of set theory, but no attempt is made to justify every statement on such a basis.
		This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union).
		Intuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other as the second element, and having the fundamental property that, two ordered pairs are equal if and only if their first elements are equal and their second elements are equal.
	www.wikipedia.org /wiki/Basic_set_theory (2554 words)

	Set union (Site not responding. Last check: 2007-10-14)
		The union of A and B is standardly written"A ∪ B".
		Binary union (the union of just two sets at a time) is an associative operation; that is, A ∪ (B ∪ C) = (A ∪ B) ∪ C.
		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 formal set theory.
	www.therfcc.org /set-union-272583.html (617 words)

	Axiom of infinity - Encyclopedia, History, Geography and Biography
		or in words: There is a set N, such that the empty set is in N and such that whenever x is a member of N, the set formed by taking the union of x with its singleton {x} is also a member of N.
		Note that the axiom of pairing allows us to form the singleton {x}, and also to form the pair.
		This set S may contain more than just the natural numbers, forming a subset of it, but we may apply the axiom schema of specification to remove unwanted elements, leaving the set N of all natural numbers.
	www.arikah.net /encyclopedia/Axiom_of_infinity (399 words)

	Axiom of pairing (Site not responding. Last check: 2007-10-14)
		In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is oneof the axioms of Zermelo-Fraenkel set theory.
		What the axiom is really saying is that, given two sets A and B, we can find a set C whose membersare precisely A and B.
		Thus, one may use this as an axiom schema in the place of the axioms ofempty set and pairing.
	www.therfcc.org /axiom-of-pairing-160924.html (462 words)

	Zermelo
		be the union of the elements of A.
		This axiom is not always used -- it seems to have no application to mathematics, but it does make some proofs and definitions easier, e.g., that of an ordinal.
		The open question of whether one could develop a set theory with a finite number of axioms was answered in the affirmative by J. von Neumann in 1925.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/zermelo/zermelo.html (1172 words)

	Read about Union (set theory) at WorldVillage Encyclopedia. Research Union (set theory) and learn about Union (set ... (Site not responding. Last check: 2007-10-14)
		mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else.
		That this union of M is a set no matter how large a set M itself might be, is the content of the
		The analogy between finite unions and logical disjunction extends to one between infinite unions and
	encyclopedia.worldvillage.com /s/b/Set_theoretic_union (634 words)

	Axiom of union (Site not responding. Last check: 2007-10-14)
		The axiom of union allows us to combine the objects in many different sets and make them members of a single new set.
		It says we can go down two levels taking not the members of a set but the members of members of a set and combine them into a new set.
		that is the union of all the members of
	www.mtnmath.com /whatth/node26.html (70 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Here are all the axioms in ZFC, one of the most popular modern mathematical theories.
		Axiom of Exstensionality: Two sets are equal iff they have the same elements.
		x e T) Axiom of Union: Given any set of sets, there is a set which is the union of all the sets in the set.
	www.math.niu.edu /~rusin/known-math/98/zfc (300 words)

	Axiom of infinity
		The integers are defined by an axiom that asserts the existence of a set
		The remaining axioms are developed in the next chapter starting in Section 6.3.
		The discussion of the infinite at the end of this chapter and the start of the next lays the groundwork for those axioms.
	www.mtnmath.com /whatr72h/node47.html (50 words)

	Axiom of union - Encyclopedia, History, Geography and Biography
		Axiom of union - Encyclopedia, History, Geography and Biography
		A as {C in B : for all D in A, C is in D} using the axiom schema of specification.
		This encyclopedia, history, geography and biography article about Axiom of union contains research on
	www.arikah.net /encyclopedia/Axiom_of_union (344 words)

	PlanetMath: union
		From an axiomatic point of view, the existence of the union is guaranteed by the axiom of union.
		This is version 3 of union, born on 2002-01-26, modified 2005-06-11.
		Object id is 1619, canonical name is Union.
	planetmath.org /encyclopedia/Union.html (63 words)

	► » axiom of union (Site not responding. Last check: 2007-10-14)
		How to prouve that the union of two sets is a set with the axiom of union?
		The axiom of union says that it is possible to consider the union of all
		Therefore, C is the union of A and B. No, no! The is no need to invoke choice here.
	www.science-chat.org /detail-6394867.html (792 words)

	Axiom of pairing (Site not responding. Last check: 2007-10-14)
		Thus the essence of the Axiom is: :Any two sets have a pair.
		Together with the axiom of empty set, the Axiom of pairing can be generalised to the following statement: :
		We can extend this schema to include n=0 if we interpret that case as the axiom of empty set.
	axiom-of-pairing.infohub.dnip.net (458 words)

	Axiom of Choice (Site not responding. Last check: 2007-10-14)
		The axiom of choice is fundamentally different from the other axioms of ZF.
		Another result of the axiom of choice is that the union of a countable collection of countable sets is countable.
		The axiom of choice occurs pretty frequently in mathematics, particularly in the form of Zorn's Lemma, known by some authors as the Kuratowski-Zorn Lemma.
	br.endernet.org /~loner/settheory/axiomofchoice/ac.html (539 words)

	Naive set theory Unions, intersections, and relative complements Paradoxes empty set Functional programming axiom of ... (Site not responding. Last check: 2007-10-14)
		For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6.If A and B are equal, then this is denoted symbolically as A = B (as usual).
		Given two sets A and B, we may construct their union.This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union).
		P(A) is a Boolean algebra under the operations of union and intersection.
	en.powerwissen.com /F0YLFogFQDeP0L3HLou3qA%3D%3D_Naive_set_theory.html (2106 words)

Try your search on: Qwika (all wikis)