Coupling constant: Difference between revisions

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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
 
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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.
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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''&nbsp;=&nbsp;{1,&nbsp;2,&nbsp;{3,&nbsp;4}<nowiki>}</nowiki>. The elements of ''B'' are ''not'' 1, 2, 3, and&nbsp;4.  Rather, there are only three elements of ''B'', namely the numbers 1 and 2, and the set {3,&nbsp;4}.
 
The elements of a set can be anything.  For example, ''C''&nbsp;=&nbsp;{&nbsp;red,&nbsp;green,&nbsp;blue&nbsp;}, 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&nbsp;"∈". Writing
 
:<math>x \in A </math>
 
means that "''x'' is an element of&nbsp;''A''". Equivalent expressions are "''x'' is a member of&nbsp;''A''", "''x'' belongs to&nbsp;''A''", "''x'' is in&nbsp;''A''" and "''x'' lies in&nbsp;''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&nbsp;''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&nbsp;"∉". Writing
:<math>x \notin A</math>
means that "''x'' is not an element of&nbsp;''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&nbsp;''A'' is&nbsp;4, while the cardinality of either of the sets ''B'' and ''C'' is&nbsp;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'''&nbsp;=&nbsp;{&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;4,&nbsp;...&nbsp;}.
 
==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''&nbsp;=&nbsp;{&nbsp;2,&nbsp;4,&nbsp;&nbsp;8,&nbsp;10,&nbsp;12&nbsp;} is finite and equal to&nbsp;5.
*The cardinality of ''P''&nbsp;=&nbsp;{&nbsp;2,&nbsp;3,&nbsp;5,&nbsp;7,&nbsp;11,&nbsp;13,&nbsp;...} (the [[prime numbers]]) is infinite&nbsp;(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]]

Latest revision as of 21:22, 8 December 2014

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