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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 Ancestral relation ： ウィキペディア英語版 Ancestral relation In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation ''R'' is its transitive closure, however defined in a different way, see below. Ancestral relations make their first appearance in Frege's ''Begriffsschrift''. Frege later employed them in his ''Grundgesetze'' as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicist foundation of arithmetic. ==Definition== The numbered propositions below are taken from his ''Begriffsschrift'' and recast in contemporary notation. A property ''P'' is called ''R''-hereditary if, whenever ''x'' is ''P'' and ''xRy'' holds, then ''y'' is also ''P'': :$(Px\; \backslash land\; xRy)\; \backslash rightarrow\; Py$ Frege defined ''b'' to be an ''R''-ancestor of ''a'', written ''aR *b'', if ''b'' has every ''R''-hereditary property that all objects ''x'' such that ''aRx'' have: :$\backslash mathbf\backslash \; \backslash vdash\; aR^$ *b \leftrightarrow \forall F (x (aRx \to Fx) \land \forall x \forall y (Fx \land xRy \to Fy) \to Fb ) The ancestral is a transitive relation: :$\backslash mathbf\backslash \; \backslash vdash\; (aR^$ *b \land bR^ *c) \rightarrow aR^ *c Let the notation ''I''(''R'') denote that ''R'' is functional (Frege calls such relations "many-one"): :$\backslash mathbf\backslash \; \backslash vdash\; I(R)\; \backslash leftrightarrow\; \backslash forall\; x\; \backslash forall\; y\; \backslash forall\; z\; (\backslash land\; xRz)\; \backslash rightarrow\; y=z)$ If ''R'' is functional, then the ancestral of ''R'' is what nowadays is called connected: :$\backslash mathbf\backslash \; \backslash vdash\; (I(R)\; \backslash land\; aR^$ *b \land aR^ *c) \rightarrow (bR^ *c \lor b=c \lor cR^ *b) 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「Ancestral 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.