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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 : is called a family of elements in ''X'' indexed by ''I'' . An indexed family can be turned into a set by considering the set , that is, the range of ''x''. However, the mapping x does not need to be injective, that is, there may exist with but . Thus, where |A| denotes the cardinality of the set. Definition. Let ''I'' and ''S'' be sets. An indexed family of sets with 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
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