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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 Fuzzy sets ： ウィキペディア英語版 Fuzzy set In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh〔L. A. Zadeh (1965) ("Fuzzy sets" ). ''Information and Control'' 8 (3) 338–353.〕 and Dieter Klaua〔Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided by 〕 in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an ''L''-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics decision-making and clustering , are special cases of ''L''-relations when ''L'' is the unit interval (1 ). In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval (). Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.〔D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.〕 In fuzzy set theory, classical bivalent sets are usually called ''crisp'' sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.〔Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, "FM-test: A Fuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis", BMC Bioinformatics, 7 (Suppl 4): S7. 2006.〕 ==Definition== A fuzzy set is a pair $(U,\; m)$ where $U$ is a set and $m\backslash colon\; U\; \backslash rightarrow\; ().$ For each $x\backslash in\; U,$ the value $m(x)$ is called the grade of membership of $x$ in $(U,m).$ For a finite set $U=\backslash ,$ the fuzzy set $(U,\; m)$ is often denoted by $\backslash .$ Let $x\; \backslash in\; U.$ Then $x$ is called not included in the fuzzy set $(U,m)$ if , $x$ is called fully included if , and $x$ is called a fuzzy member if The set $\backslash$ is called the support of $(U,m)$ and the set $\backslash$ is called its kernel or core. The function $m$ is called the membership function of the fuzzy set $(U,\; m).$ Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure $L$ of a given kind; usually it is required that $L$ be at least a poset or lattice. These are usually called ''L''-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in () are then called ()-valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.〔Goguen, Joseph A., 196, "''L''-fuzzy sets". ''Journal of Mathematical Analysis and Applications'' 18: 145–174〕 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「Fuzzy 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.