Imaginary unit - Revision history

en>JohnBlackburne: Reverted 1 edit by Dash77 (talk): Again, no need to introduce complex numbers, a more advanced topic. It's clearer just considering square roots of real numbers. (TW)

2014-12-15T03:08:08Z

Reverted 1 edit by Dash77 (talk): Again, no need to introduce complex numbers, a more advanced topic. It's clearer just considering square roots of real numbers. (TW)

← Older revision		Revision as of 05:08, 15 December 2014
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en>Tamfang: /* top */ +link

2014-02-22T05:57:13Z

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en>Tamfang: /* Alternative notations */ 'j' on its own has no predefined meaning in Python

2014-01-21T02:38:09Z

Alternative notations: 'j' on its own has no predefined meaning in Python

← Older revision		Revision as of 04:38, 21 January 2014
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	I'~~m Sidney~~ (18) ~~from Unterwuhr~~, ~~Austria~~. <br>~~I'm learning Russian literature at~~ a ~~local high school~~ and I'm just about to ~~graduate~~.<br>~~I have~~ a ~~part time job~~ in a ~~backery~~.<br><br>~~My blog post;~~ [~~http:~~//~~johnjesensky8190~~.~~wordpress.com~~/~~2014~~/08/25/get-~~assist~~-~~from~~-a-~~terry~~-~~bandy~~-~~estate~~-~~preparing~~-~~lawyer~~-to-~~create~~-a-~~valid~~-~~will/ terry bandy~~]		{{no footnotes\|date=March 2013}}
			In [[axiomatic set theory]] and the branches of [[logic]], [[mathematics]], and [[computer science]] that use it, the '''axiom of pairing''' is one of the [[axiom]]s of [[Zermelo–Fraenkel set theory]].

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

			or in simpler words:
			:Given two sets, there is a set whose members are exactly the two given sets.

			== Interpretation ==

			What the axiom is really saying is that, given two sets ''A'' and ''B'', we can find a set ''C'' whose members are precisely ''A'' and ''B''.
			We can use the [[axiom of extensionality]] to show that this set ''C'' is unique.
			We call the set ''C'' the ''pair'' of ''A'' and ''B'', and denote it {''A'',''B''}.
			Thus the essence of the axiom is:
			:Any two sets have a pair.
			{''A'',''A''} is abbreviated {''A''}, called the ''[[singleton (mathematics)\|singleton]]'' containing ''A''.
			Note that a singleton is a special case of a pair.

			The axiom of pairing also allows for the definition of [[ordered pairs]]. For any sets <math>a</math> and <math>b</math>, the [[ordered pair]] is defined by the following:

			:<math> (a, b) = \{ \{ a \}, \{ a, b \} \}.\,</math>

			Note that this definition satisfies the condition

			:<math>(a, b) = (c, d) \iff a = c \and b = d. </math>

			Ordered [[tuple\|''n''-tuples]] can be defined recursively as follows:

			:<math> (a_1, \ldots, a_n) = ((a_1, \ldots, a_{n-1}), a_n).\!</math>

			== Non-independence ==
			The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative [[axiomatization]] of set theory. Nevertheless, in the standard formulation of the [[Zermelo–Fraenkel set theory]], the axiom of pairing follows from the [[axiom schema of replacement]] applied to any given set with two or more elements, and thus it is sometimes omitted. The existence of such a set with two elements, such as { {}, { {} } }, can be deduced either from the [[axiom of empty set]] and the [[axiom of power set]] or from the [[axiom of infinity]].

			== Generalisation ==

			Together with the [[axiom of empty set]], the axiom of pairing can be generalised to the following schema:
			:<math>\forall A_1 \, \ldots \, \forall A_n \, \exist C \, \forall D \, [D \in C \iff (D = A_1 \or \cdots \or D = A_n)]</math>
			that is:
			:Given any [[finite set\|finite]] number of sets ''A''<sub>1</sub> through ''A''<sub>''n''</sub>, there is a set ''C'' whose members are precisely ''A''<sub>1</sub> through ''A''<sub>''n''</sub>.
			This set ''C'' is again unique by the axiom of extension, and is denoted {''A''<sub>1</sub>,...,''A''<sub>''n''</sub>}.

			Of course, we can't refer to a ''finite'' number of sets rigorously without already having in our hands a (finite) set to which the sets in question belong.
			Thus, this is not a single statement but instead a [[schema (logic)\|schema]], with a separate statement for each [[natural number]] ''n''.
			*The case ''n'' = 1 is the axiom of pairing with ''A'' = ''A''<sub>1</sub> and ''B'' = ''A''<sub>1</sub>.
			*The case ''n'' = 2 is the axiom of pairing with ''A'' = ''A''<sub>1</sub> and ''B'' = ''A''<sub>2</sub>.
			*The cases ''n'' > 2 can be proved using the axiom of pairing and the [[axiom of union]] multiple times.
			For example, to prove the case ''n'' = 3, use the axiom of pairing three times, to produce the pair {''A''<sub>1</sub>,''A''<sub>2</sub>}, the singleton {''A''<sub>3</sub>}, and then the pair {{''A''<sub>1</sub>,''A''<sub>2</sub>},{''A''<sub>3</sub>}}.
			The axiom of union then produces the desired result, {''A''<sub>1</sub>,''A''<sub>2</sub>,''A''<sub>3</sub>}. We can extend this schema to include ''n''=0 if we interpret that case as the [[axiom of empty set]].

			Thus, one may use this as an [[axiom schema]] in the place of the axioms of empty set and pairing. Normally, however, one uses the axioms of empty set and pairing separately, and then proves this as a [[theorem]] schema. Note that adopting this as an axiom schema will not replace the [[axiom of union]], which is still needed for other situations.

			==Another alternative==
			Another axiom which implies the axiom of pairing in the presence of the [[axiom of empty set]] is
			:<math>\forall A \, \forall B \, \exist C \, \forall D \, [D \in C \iff (D \in A \or D = B)]</math>.
			Using {} for ''A'' and ''x'' for B, we get {''x''} for C. Then use {''x''} for ''A'' and ''y'' for ''B'', getting {''x,y''} for C. One may continue in this fashion to build up any finite set. And this could be used to generate all [[hereditarily finite set]]s without using the [[axiom of union]].

			== 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).
			*Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
			*Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

			{{Set theory}}

			[[Category:Axioms of set theory]]

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

en>DVdm: Reverted edits by 2602:306:2562:28C9:F9:264D:4D02:97D2 (talk) to last version by Loadmaster

2012-08-31T06:59:05Z

Reverted edits by 2602:306:2562:28C9:F9:264D:4D02:97D2 (talk) to last version by Loadmaster

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I'm Sidney (18) from Unterwuhr, Austria. <br>I'm learning Russian literature at a local high school and I'm just about to graduate.<br>I have a part time job in a backery.<br><br>My blog post; [http://johnjesensky8190.wordpress.com/2014/08/25/get-assist-from-a-terry-bandy-estate-preparing-lawyer-to-create-a-valid-will/ terry bandy]