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In [[non-standard analysis]], a branch of [[mathematics]], a '''hyperfinite set''' or '''*-finite set''' is a type of [[internal set]]. An internal set ''H'' of internal cardinality ''g'' ∈ *'''N''' (the [[hypernatural]]s) is hyperfinite [[if and only if]] there exists an internal [[bijection]] between ''G'' = {1,2,3,...,''g''} and ''H''.<ref>{{cite book|title=Optimization and nonstandard analysis|author=J. E. Rubio|publisher=Marcel Dekker|year=1994|isbn=0-8247-9281-5|page=110}}</ref><ref name=Chuaqui /> Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *'''R''' always exists, leading to the possibility of well-defined [[integration (mathematics)|integration]].<ref name=Chuaqui>{{cite book|title=Truth, possibility, and probability: new logical foundations of probability and statistical inference|author=R. Chuaqui|publisher=Elsevier|year=1991|isbn=0-444-88840-3|pages=182–3}}</ref> | |||
Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a ''near interval'' with respect to that interval. Consider a hyperfinite set <math>K = {k_1,k_2, \dots ,k_n}</math> with a hypernatural ''n''. ''K'' is a near interval for [''a'',''b''] if ''k''<sub>1</sub> = ''a'' and ''k''<sub>''n''</sub> = ''b'', and if the difference between successive elements of ''K'' is [[infinitesimal]]. Phrased otherwise, the requirement is that for every ''r'' ∈ [''a'',''b''] there is a ''k''<sub>''i''</sub> ∈ ''K'' such that ''k''<sub>''i''</sub> ≈ ''r''. This, for example, allows for an approximation to the [[unit circle]], considered as the set <math>e^{i\theta}</math> for θ in the interval [0,2π].<ref name=Chuaqui /> | |||
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.<ref>{{cite book|title=Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory|author=[[Luigi Ambrosio|L. Ambrosio]] et al.|publisher=Springer|year=2000|isbn=3-540-64803-8|page=203}}</ref> | |||
== Ultrapower construction == | |||
In terms of the [[ultrapower]] construction, the hyperreal line *'''R''' is defined as the collection of [[equivalence class]]es of sequences <math>\langle u_n, n=1,2,\ldots \rangle</math> of real numbers ''u''<sub>''n''</sub>. Namely, the equivalence class defines a hyperreal, denoted <math>[u_n]</math> in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *'''R''' is of the form <math>[A_n]</math>, and is defined by a sequence <math>\langle A_n \rangle</math> of finite sets <math>A_n \subset \mathbb{R}, n=1,2,\ldots</math><ref>{{cite book|author=R. Goldblatt|year=1998|title=Lectures on the hyperreals. An introduction to nonstandard analysis|page=188|publisher=Springer|isbn=0-387-98464-X}}</ref> | |||
== Notes == | |||
<references/> | |||
== External links == | |||
*{{mathworld |urlname=HyperfiniteSet |title=Hyperfinite Set |author=M. Insall}} | |||
{{Infinitesimals}} | |||
{{DEFAULTSORT:Hyperfinite Set}} | |||
[[Category:Non-standard analysis]] | |||
Revision as of 09:37, 15 December 2013
In non-standard analysis, a branch of mathematics, a hyperfinite set or *-finite set is a type of internal set. An internal set H of internal cardinality g ∈ *N (the hypernaturals) is hyperfinite if and only if there exists an internal bijection between G = {1,2,3,...,g} and H.[1][2] Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *R always exists, leading to the possibility of well-defined integration.[2]
Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a near interval with respect to that interval. Consider a hyperfinite set with a hypernatural n. K is a near interval for [a,b] if k1 = a and kn = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ [a,b] there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set for θ in the interval [0,2π].[2]
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.[3]
Ultrapower construction
In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences of real numbers un. Namely, the equivalence class defines a hyperreal, denoted in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form , and is defined by a sequence of finite sets [4]
![{\displaystyle [u_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23be03a47ad073f04989357c7af10c7daba90798)
![{\displaystyle [A_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c33cf5efdb1084af6560e718afd9151903fcd467)
Notes
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 2.0 2.1 2.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
Here is my web site - cottagehillchurch.com