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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 axiom of extensionality ： ウィキペディア英語版 axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. == Formal statement == In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :$\backslash forall\; A\; \backslash ,\; \backslash forall\; B\; \backslash ,\; (\; \backslash forall\; X\; \backslash ,\; (X\; \backslash in\; A\; \backslash iff\; X\; \backslash in\; B)\; \backslash Rightarrow\; A\; =\; B)$ or in words: :Given any set ''A'' and any set ''B'', if for every set ''X'', ''X'' is a member of ''A'' if and only if ''X'' is a member of ''B'', then ''A'' is equal to ''B''. :(It is not really essential that ''X'' here be a ''set'' — but in ZF, everything is. See Ur-elements below for when this is violated.) The converse, $\backslash forall\; A\; \backslash ,\; \backslash forall\; B\; \backslash ,\; (A\; =\; B\; \backslash Rightarrow\; \backslash forall\; X\; \backslash ,\; (X\; \backslash in\; A\; \backslash iff\; X\; \backslash in\; B)\; )$, of this axiom follows from the substitution property of equality. 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「axiom of extensionality」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.