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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 Indexed family ： ウィキペディア英語版 Indexed family In mathematics, an indexed family is a collection of values associated with indices. For example, a ''family of real numbers, indexed by the integers'' is a collection of real numbers, where each integer is associated with one of the real numbers. Formally, an indexed family is the same thing as a mathematical function; a function with domain ''J'' and codomain ''X'' is equivalent to a family of elements of ''X'' indexed by elements of ''J''. The difference is conceptual; indexed families are interpreted as collections instead of as functions. Every element of the image of the family's underlying function is an element of the family. When a function ''f'' : ''J'' → ''X'' is treated as a family, ''J'' is called the ''index set'' of the family, the function image ''f''(''j'') for ''j'' ∈ ''J'' is denoted ''x''''j'', and the mapping ''f'' is denoted ''j''∈''J'' or simply . Next, if the set ''X'' is the power set of a set ''U'', then the family ''j''∈''J'' is called a family of sets indexed by ''J'' . ==Mathematical statement== Definition. Let ''I'' and ''X'' be sets. The function :$\backslash begin$ x\colon I &\to X \\ i &\mapsto x_i = x(i) \end is called a family of elements in ''X'' indexed by ''I'' . An indexed family can be turned into a set by considering the set $\backslash mathcal\; :=\; \backslash$, that is, the range of ''x''. However, the mapping x does not need to be injective, that is, there may exist $i,j\; \backslash in\; I$ with $i\; \backslash neq\; j$ but $x\_i\; =\; x\_j$. Thus, $|\; \backslash mathcal|$ \leq |\_ | where |A| denotes the cardinality of the set. Definition. Let ''I'' and ''S'' be sets. An indexed family of sets $\backslash \_$ with $C\_i\; \backslash subset\; S$ is an indexed family that maps elements of the index set ''I'' to elements of the power set of ''S''. Hence, an indexed family of sets is conceptually different from a family of sets (which is just a synonym for "set of sets"), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family. 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「Indexed family」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.