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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 Foundational relation ： ウィキペディア英語版 Foundational relation In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimal element. Formally, let (''A'', ''R'') be a binary relation structure, where ''A'' is a class (set or proper class), and ''R'' is a binary relation defined on ''A''. Then (''A'', ''R'') is a foundational relation if and only if any nonempty subset in ''A'' has a ''R''-minimal element. In predicate logic, :$(\backslash forall\; S)\backslash left(S\; \backslash subseteq\; A\; \backslash wedge\; S\; \backslash not=\; \backslash emptyset\; \backslash Rightarrow\; (\backslash exists\; x\; \backslash in\; S)(S\; \backslash cap\; R^\backslash \; =\; \backslash emptyset)\backslash right),$ 〔See Definition 6.21 in 〕 in which ∅ denotes the empty set, and ''R''−1 denotes the class of the elements that precede ''x'' in the relation ''R''. That is, :$R^\backslash \; =\; \backslash .$ 〔See Theorem 6.19 and Definition 6.20 in 〕 Here ''x'' is an ''R''-minimal element in the subset ''S'', since none of its ''R''-predecessors is in ''S''. ==See also== * Binary relation * Well-order 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「Foundational relation」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.