翻訳と辞書 Words near each other ・ Axl Smith ・ Axle ・ Axle counter ・ Axle hitch ・ Axiom of countable choice ・ Axiom of dependent choice ・ Axiom of determinacy ・ Axiom of empty set ・ Axiom of Equity ・ Axiom of extensionality ・ Axiom of global choice ・ Axiom of infinity ・ Axiom of limitation of size ・ Axiom of Maria ・ Axiom of pairing ・ Axiom of power set ・ Axiom of projective determinacy ・ Axiom of real determinacy ・ Axiom of reducibility ・ Axiom of regularity ・ Axiom of union ・ Axiom S5 ・ Axiom schema ・ Axiom schema of predicative separation ・ Axiom schema of replacement ・ Axiom schema of specification ・ Axiom Telecom ・ Axiom Verge ・ Axiom-man ・ Axioma Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

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 power set ： ウィキペディア英語版 Axiom of power set In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :$\backslash forall\; A\; \backslash ,\; \backslash exists\; P\; \backslash ,\; \backslash forall\; B\; \backslash ,\; (\backslash in\; P\; \backslash iff\; \backslash forall\; C\; \backslash ,\; (C\; \backslash in\; B\; \backslash Rightarrow\; C\; \backslash in\; A))$ where ''P'' stands for the Power set of ''A'', $\backslash mathcal(A)$. In English, this says: :Given any set ''A'', there is a set $\backslash mathcal(A)$ such that, given any set ''B'', ''B'' is a member of $\backslash mathcal(A)$ if and only if every element of ''B'' is also an element of ''A''. More succinctly: ''for every set $A$, there is a set $\backslash mathcal(A)$ consisting precisely of the subsets of $A$.'' Note the subset relation $\backslash subseteq$ is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, $\backslash in$. By the axiom of extensionality, the set $\backslash mathcal(A)$ is unique. The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity. == Consequences == The Power Set Axiom allows a simple definition of the Cartesian product of two sets $X$ and $Y$: :$X\; \backslash times\; Y\; =\; \backslash .$ Notice that :$x,\; y\; \backslash in\; X\; \backslash cup\; Y$ :$\backslash ,\; \backslash \; \backslash in\; \backslash mathcal(X\; \backslash cup\; Y)$ :$(x,\; y)\; =\; \backslash \; \backslash \}\; \backslash in\; \backslash mathcal(\backslash mathcal(X\; \backslash cup\; Y))$ and thus the Cartesian product is a set since :$X\; \backslash times\; Y\; \backslash subseteq\; \backslash mathcal(\backslash mathcal(X\; \backslash cup\; Y)).$ One may define the Cartesian product of any finite collection of sets recursively: :$X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_n\; =\; (X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_)\; \backslash times\; X\_n.$ Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory. 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 power set」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.