Axiom of empty set - Revision history

en>Wavelength: correcting 1 misplaced modifier: "That said" —> "However"—http://grammar.ccc.commnet.edu/grammar/modifiers.htm

2014-12-11T02:39:28Z

correcting 1 misplaced modifier: "That said" —> "However"—http://grammar.ccc.commnet.edu/grammar/modifiers.htm

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	~~In [[axiomatic set theory]]~~ and the ~~branches of [[logic]]~~, ~~[[mathematics]]~~, ~~and [[computer science]] that use~~ it, the ~~'''axiom~~ of ~~union''' is~~ one of ~~the [[axiom]]s 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''~~. ~~Together with~~ the ~~[[axiom~~ of ~~pairing]] this implies that for any two sets~~, ~~there is a set that contains exactly~~ the ~~elements~~ of ~~both~~.

	~~== Formal statement ==~~
	~~In the [[formal language]]~~ of ~~the Zermelo–Fraenkel axioms~~, ~~the axiom reads:~~
	~~:<math>\forall A\~~, ~~\exist B\~~, ~~\forall c\~~, ~~(c \in B \iff \exist D\~~, ~~(c \in D \~~and ~~D \in A)\,)~~<~~/math~~>
	~~or in words~~:
	:~~[[Given any]] [[Set (mathematics)\|set]] ''A''~~, ~~[[Existential quantification\|there is]]~~ a ~~set ''B'' such that, for any element ''c'', ''c''~~ is ~~a member of ''B'' [[if~~ and ~~only if]] there~~ is ~~a set ''D'' such that ''c''~~ is ~~a member~~ of ~~''D'' [[logical conjunction\|and]] ''D'' is~~ a ~~member~~ of ~~''A''.~~

	~~== Interpretation ==~~

	~~What~~ the ~~axiom is really saying is~~ that, ~~given a set ''A''~~, ~~we can find a set ''B'' whose members~~ are ~~precisely the members of the members of ''A''~~. ~~By the [[axiom of extensionality]] this set ''B'' is unique and it is called the ''[[union (set theory)\|union]]'' of ''A''~~, ~~and denoted~~ <~~math~~>~~\bigcup A~~<~~/math~~>. ~~Thus~~ the ~~essence~~ of the ~~axiom is:~~
	~~:The union of a set is a set~~.

	~~The axiom~~ of ~~union is generally considered uncontroversial~~, and ~~it or an equivalent appears~~ in ~~just about any alternative [[axiomatization]] of set theory~~.

	~~Note that there is no corresponding axiom of [[intersection (set theory)\|intersection]]~~. ~~If ''A'' is a ''nonempty'' set containing ''E'', then we can form the intersection~~ <~~math~~>~~\bigcap A~~<~~/math~~> ~~using the [[axiom schema~~ of ~~specification]] as~~
	~~:{''c'' in ''E'': for all ''D'' in ''A'', ''c''~~ is ~~in ''D''},~~
	~~so no separate axiom~~ of ~~intersection is necessary~~. ~~(If ''A''~~ is ~~the [[empty set]], then trying to form the intersection~~ of ~~''A'' as~~
	~~:{''c'': for all ''D'' in ''A''~~, ~~''c''~~ is ~~in ''D''}~~
	~~is not permitted by~~ the ~~axioms~~. ~~Moreover, if such a set existed, then it would contain every set~~ in the ~~"universe", but the notion of~~ a ~~[[universal set]]~~ is ~~antithetical~~ to ~~Zermelo–Fraenkel set theory~~.)

	~~== References ==~~
	*[[Paul Halmos]], ''Naive set theory''. ~~Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).~~
	*[[~~Thomas Jech\|Jech, Thomas]], 2003. ''Set Theory~~: ~~The Third Millennium Edition, Revised and Expanded''~~. ~~Springer~~. ~~ISBN 3-540-44085-2.~~
	*[[Kenneth Kunen\|Kunen, Kenneth]~~], 1980. ''Set Theory: An Introduction~~ to ~~Independence Proofs''. Elsevier. ISBN 0-444-86839-9~~.

	~~==External links==~~
	*{{planetmath reference\|id=4394\|title=Axiom of Union}}

	~~{{Set theory}}~~

	~~[[Category:Axioms of set theory]]~~

	~~[[de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC]]~~

en>Klemen Kocjancic: Added {{no footnotes}} tag to article (TW)

2013-03-09T09:39:43Z

Added {{no footnotes}} tag to article (TW)

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	~~Hello~~. ~~Let me introduce~~ the ~~author~~. ~~Her name~~ is ~~Emilia Shroyer but it~~'~~s not~~ the ~~most female name out there~~. ~~To play baseball~~ is the ~~pastime he will by no means stop performing~~. ~~California~~ is ~~our beginning location~~. ~~Managing people~~ is ~~what I do~~ and the ~~wage has been truly satisfying.~~<br><br>~~my page~~ :: [~~https~~:~~//www~~.~~tumblr~~.~~com/blog/mmattchuu https~~:~~//www~~.~~tumblr~~.~~com/~~]		{{no footnotes\|date=March 2013}}
			In [[axiomatic set theory]] and the branches of [[logic]], [[mathematics]], and [[computer science]] that use it, the '''axiom of union''' is one of the [[axiom]]s 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''. Together with the [[axiom of pairing]] this implies that for any two sets, there is a set that contains exactly the elements of both.

			== Formal statement ==
			In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:
			:<math>\forall A\, \exist B\, \forall c\, (c \in B \iff \exist D\, (c \in D \and D \in A)\,)</math>
			or in words:
			:[[Given any]] [[Set (mathematics)\|set]] ''A'', [[Existential quantification\|there is]] a set ''B'' such that, for any element ''c'', ''c'' is a member of ''B'' [[if and only if]] there is a set ''D'' such that ''c'' is a member of ''D'' [[logical conjunction\|and]] ''D'' is a member of ''A''.

			== Interpretation ==

			What the axiom is really saying is that, given a set ''A'', we can find a set ''B'' whose members are precisely the members of the members of ''A''. By the [[axiom of extensionality]] this set ''B'' is unique and it is called the ''[[union (set theory)\|union]]'' of ''A'', and denoted <math>\bigcup A</math>. Thus the essence of the axiom is:
			:The union of a set is a set.

			The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative [[axiomatization]] of set theory.

			Note that there is no corresponding axiom of [[intersection (set theory)\|intersection]]. If ''A'' is a ''nonempty'' set containing ''E'', then we can form the intersection <math>\bigcap A</math> using the [[axiom schema of specification]] as
			:{''c'' in ''E'': for all ''D'' in ''A'', ''c'' is in ''D''},
			so no separate axiom of intersection is necessary. (If ''A'' is the [[empty set]], then trying to form the intersection of ''A'' as
			:{''c'': for all ''D'' in ''A'', ''c'' is in ''D''}
			is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of a [[universal set]] is antithetical to Zermelo–Fraenkel set theory.)

			== References ==
			*[[Paul Halmos]], ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
			*[[Thomas Jech\|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
			*[[Kenneth Kunen\|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

			==External links==
			*{{planetmath reference\|id=4394\|title=Axiom of Union}}

			{{Set theory}}

			[[Category:Axioms of set theory]]

			[[de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC]]

42.2.52.16: /* References */

2012-06-22T02:17:45Z

References

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Hello. Let me introduce the author. Her name is Emilia Shroyer but it's not the most female name out there. To play baseball is the pastime he will by no means stop performing. California is our beginning location. Managing people is what I do and the wage has been truly satisfying.<br><br>my page :: [https://www.tumblr.com/blog/mmattchuu https://www.tumblr.com/]