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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hyperfinite set ： ウィキペディア英語版 Hyperfinite set 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'' = and ''H''.〔 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. 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 $K\; =$ with a hypernatural ''n''. ''K'' is a near interval for () if ''k''1 = ''a'' and ''k''''n'' = ''b'', and if the difference between successive elements of ''K'' is infinitesimal. Phrased otherwise, the requirement is that for every ''r'' ∈ () there is a ''k''''i'' ∈ ''K'' such that ''k''''i'' ≈ ''r''. This, for example, allows for an approximation to the unit circle, considered as the set $e^$ for θ in the interval ().〔 In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set. == Ultrapower construction == In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences $\backslash langle\; u\_n,\; n=1,2,\backslash ldots\; \backslash rangle$ of real numbers ''u''''n''. 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 $\backslash langle\; A\_n\; \backslash rangle$ of finite sets $A\_n\; \backslash subset\; \backslash mathbb,\; n=1,2,\backslash ldots$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hyperfinite set」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.