Coupling constant - Revision history

en>Feon: made "Beta functions", "QED and the Landau pole", "QCD and asymptotic freedom" and "QCD scale" sections a subsections of "Running coupling" because they are the subtopics of the running coupling, as opposed to scale-independent couplings

2014-12-08T19:22:32Z

made "Beta functions", "QED and the Landau pole", "QCD and asymptotic freedom" and "QCD scale" sections a subsections of "Running coupling" because they are the subtopics of the running coupling, as opposed to scale-independent couplings

← Older revision		Revision as of 21:22, 8 December 2014
Line 1:		Line 1:
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	~~== Sets ==~~
	~~Writing ''A'' = {1, 2, 3, 4 } means that the elements of the set ''A'' are the numbers 1, 2,~~ 3 ~~and 4. Sets of elements of ''A'', for example {1~~,~~ 2}, are~~ [~~[subset]]s of ''A''~~.

	~~Sets can themselves be elements~~. ~~For example consider the set ''B'' = {1, 2, {3, 4}<nowiki>}<~~/~~nowiki>. The elements of ''B'' are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of ''B'', namely~~ the ~~numbers 1 and 2, and~~ the ~~set {3, 4}.~~

	~~The elements of a set can be anything. For example, ''C'' = { red, green, blue }~~, ~~is the set whose elements are the colors red, green and blue.~~

	~~==Notation and terminology==~~
	~~The~~ [~~[Relation (mathematics)\|relation]] "is an element of", also called '''set membership''', is denoted by the symbol "∈". Writing~~

	:~~<math>x \in A <~~/~~math>~~

	~~means that "''x'' is an element of ''A''"~~. ~~Equivalent expressions are "''x'' is a member of ''A''", "''x'' belongs to ''A''", "''x'' is in ''A''" and "''x'' lies in ''A''"~~. The expressions "''A'' includes ''x''" and "''A'' contains ''x''" are also used to mean set membership, however some authors use them to mean instead "''x'' is a [[subset]] of ''A''".<ref name="schech">{{cite book \|author = [[Eric Schechter]] \|title= Handbook of Analysis and Its Foundations \|publisher= [[Academic Press]] \|year= 1997\|isbn= 0-12-~~622760~~-~~8 }} p. 12<~~/~~ref> Logician~~ [~~[George Boolos]] strongly urged that "contains" be used for membership only and "includes" for the subset relation only~~.~~<ref name="boolos">{{cite speech~~
	~~\| title = 24~~.~~243 Classical Set Theory (lecture).~~
	~~\| author = [[George Boolos]]~~
	~~\| authorlink =~~
	~~\| date = February 4, 1992~~
	~~\| location = Massachusetts Institute of Technology, Cambridge, MA~~
	~~\| url =~~
	~~\| accessdate =~~
	~~}}<~~/~~ref>~~

	~~Another possible notation for the same relation is~~

	~~:<math>A \ni x,<~~/~~math>~~

	~~meaning "''A'' contains ''x''", though it is used less often.~~

	~~The [[negation]~~] of ~~set membership is denoted~~ by the ~~symbol "∉". Writing~~
	~~:<math>x \notin A</math>~~
	~~means that "''x'' is not~~ an ~~element~~ of~~ ''A''"~~.

	~~The~~ [~~[Unicode]] characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209 ('not an element of')~~. ~~The equivalent [[LaTeX]] commands are "\in", "\ni" and "\notin"~~. ~~[[Mathematica]] has commands "\[Element~~]~~" and "\[NotElement]".~~

	~~==Cardinality of sets==~~
	~~{{main\|Cardinality}}~~
	~~The number of elements in a particular set~~ is a ~~property known as [[cardinality]], informally this is~~ the ~~size of a set. In~~ the ~~above examples~~ the ~~cardinality of the set ''A'' is 4, while~~ the ~~cardinality of either of~~ the ~~sets ''B''~~ and ~~''C'' is 3. An infinite set is~~ a ~~set with an infinite number of elements, while~~ a ~~finite set is~~ a ~~set~~ with ~~a finite number of elements. The above examples are examples of finite sets. An example of an infinite set~~ is the ~~set of~~ [~~[natural numbers]], '''N''' = { 1, 2, 3, 4, ... }.~~

	~~==Examples==~~
	~~Using the sets defined above, namely ''A'' = {1, 2, 3, 4 }, ''B'' = {1, 2, {3, 4}} and ''C'' = { red, green, blue }~~:
	*2 ∈ ''A''
	*{3,4} ∈ ''B''
	*{3,4} is a member of ''B''
	*Yellow ∉ ''C''
	*The cardinality of ''D'' = { 2, 4,  8, 10, 12 } is finite and equal to 5.
	*The cardinality of ''P'' = { 2, 3, 5, 7, 11, 13, ..~~.} (the [[prime numbers]]) is infinite (this was proven by [[Euclid]]).~~

	~~== References ==~~
	~~{{reflist}}~~

	~~==Further reading==~~
	*{{Citation\|last=Halmos\|first=Paul R.\|author-~~link=Paul R. Halmos\|origyear=1960\|year=1974\|title=Naive Set Theory\|publisher=Springer~~-~~Verlag\|location=NY\|edition=Hardcover\|series=Undergraduate Texts in Mathematics\|isbn=0~~-~~387~~-~~90092~~-~~6}}~~ - ~~"Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither)~~.
	*{{Citation\|last=Jech\|first=Thomas\|author-link=Thomas Jech\|year=2002\|title=Stanford Encyclopedia of ~~Philosophy\|chapter=Set Theory\|url=~~http://~~plato~~.~~stanford~~.~~edu~~/~~entries/set~~-~~theory~~/}}
	*{{Citation\|last=Suppes\|first=Patrick\|author-~~link=Patrick Suppes\|origyear=1960\|year=1972\|title=Axiomatic Set Theory\|publisher=Dover Publications, Inc.\|location=NY\|isbn=0~~-~~486~~-~~61630~~-4~~}} - Both the notion of set (~~a ~~collection of members), membership~~ or ~~element~~-~~hood~~, ~~the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed~~ for ~~a more thorough understanding of "set element".~~

	~~== External links ==~~
	*{{MathWorld \|title=Element \|id=Element }}

	~~{{logic}}~~
	~~{{Set theory}}~~

	~~[[Category:Basic concepts in set theory]]~~

	~~[[ru:Множество#Элемент множества]]~~

en>AnomieBOT: Dating maintenance tags: {{Why}}

2013-12-21T00:52:47Z

Dating maintenance tags: {{Why}}

← Older revision		Revision as of 02:52, 21 December 2013
Line 1:		Line 1:
	~~Greetings! I am Myrtle Shroyer~~. For a ~~whilst I~~'~~ve been~~ in ~~South Dakota~~ and ~~my mothers~~ and ~~fathers live nearby. He~~ used to ~~be unemployed but now he~~ is a ~~computer operator but his marketing by no means arrives~~. ~~Body building is 1~~ of the ~~issues I adore most~~.<br><br>~~Stop~~ by ~~my web blog~~ :: [http://~~wales~~.~~tomatovr~~.~~com~~/xe/~~index~~.~~php?document_srl~~=~~1500670&mid~~=~~walse05 wales~~.~~tomatovr.com~~]		In [[mathematics]], an '''element''', or '''member''', of a [[Set (mathematics)\|set]] is any one of the distinct [[Mathematical object\|objects]] that make up that set.

			== Sets ==
			Writing ''A'' = {1, 2, 3, 4 } means that the elements of the set ''A'' are the numbers 1, 2, 3 and 4. Sets of elements of ''A'', for example {1, 2}, are [[subset]]s of ''A''.

			Sets can themselves be elements. For example consider the set ''B'' = {1, 2, {3, 4}<nowiki>}</nowiki>. The elements of ''B'' are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of ''B'', namely the numbers 1 and 2, and the set {3, 4}.

			The elements of a set can be anything. For example, ''C'' = { red, green, blue }, is the set whose elements are the colors red, green and blue.

			==Notation and terminology==
			The [[Relation (mathematics)\|relation]] "is an element of", also called '''set membership''', is denoted by the symbol "∈". Writing

			:<math>x \in A </math>

			means that "''x'' is an element of ''A''". Equivalent expressions are "''x'' is a member of ''A''", "''x'' belongs to ''A''", "''x'' is in ''A''" and "''x'' lies in ''A''". The expressions "''A'' includes ''x''" and "''A'' contains ''x''" are also used to mean set membership, however some authors use them to mean instead "''x'' is a [[subset]] of ''A''".<ref name="schech">{{cite book \|author = [[Eric Schechter]] \|title= Handbook of Analysis and Its Foundations \|publisher= [[Academic Press]] \|year= 1997\|isbn= 0-12-622760-8 }} p. 12</ref> Logician [[George Boolos]] strongly urged that "contains" be used for membership only and "includes" for the subset relation only.<ref name="boolos">{{cite speech
			\| title = 24.243 Classical Set Theory (lecture).
			\| author = [[George Boolos]]
			\| authorlink =
			\| date = February 4, 1992
			\| location = Massachusetts Institute of Technology, Cambridge, MA
			\| url =
			\| accessdate =
			}}</ref>

			Another possible notation for the same relation is

			:<math>A \ni x,</math>

			meaning "''A'' contains ''x''", though it is used less often.

			The [[negation]] of set membership is denoted by the symbol "∉". Writing
			:<math>x \notin A</math>
			means that "''x'' is not an element of ''A''".

			The [[Unicode]] characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209 ('not an element of'). The equivalent [[LaTeX]] commands are "\in", "\ni" and "\notin". [[Mathematica]] has commands "\[Element]" and "\[NotElement]".

			==Cardinality of sets==
			{{main\|Cardinality}}
			The number of elements in a particular set is a property known as [[cardinality]], informally this is the size of a set. In the above examples the cardinality of the set ''A'' is 4, while the cardinality of either of the sets ''B'' and ''C'' is 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of [[natural numbers]], '''N''' = { 1, 2, 3, 4, ... }.

			==Examples==
			Using the sets defined above, namely ''A'' = {1, 2, 3, 4 }, ''B'' = {1, 2, {3, 4}} and ''C'' = { red, green, blue }:
			*2 ∈ ''A''
			*{3,4} ∈ ''B''
			*{3,4} is a member of ''B''
			*Yellow ∉ ''C''
			*The cardinality of ''D'' = { 2, 4,  8, 10, 12 } is finite and equal to 5.
			*The cardinality of ''P'' = { 2, 3, 5, 7, 11, 13, ...} (the [[prime numbers]]) is infinite (this was proven by [[Euclid]]).

			== References ==
			{{reflist}}

			==Further reading==
			*{{Citation\|last=Halmos\|first=Paul R.\|author-link=Paul R. Halmos\|origyear=1960\|year=1974\|title=Naive Set Theory\|publisher=Springer-Verlag\|location=NY\|edition=Hardcover\|series=Undergraduate Texts in Mathematics\|isbn=0-387-90092-6}} - "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
			*{{Citation\|last=Jech\|first=Thomas\|author-link=Thomas Jech\|year=2002\|title=Stanford Encyclopedia of Philosophy\|chapter=Set Theory\|url=http://plato.stanford.edu/entries/set-theory/}}
			*{{Citation\|last=Suppes\|first=Patrick\|author-link=Patrick Suppes\|origyear=1960\|year=1972\|title=Axiomatic Set Theory\|publisher=Dover Publications, Inc.\|location=NY\|isbn=0-486-61630-4}} - Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".

			== External links ==
			*{{MathWorld \|title=Element \|id=Element }}

			{{logic}}
			{{Set theory}}

			[[Category:Basic concepts in set theory]]

			[[ru:Множество#Элемент множества]]

en>EmausBot: r2.7.2+) (Robot: Modifying ko:결합 상수

2012-08-14T22:01:26Z

r2.7.2+) (Robot: Modifying ko:결합 상수

New page

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