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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 ring of sets ： ウィキペディア英語版 ring of sets In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets $\backslash mathcal$ is called a ring (of sets) if it is closed under intersection and union. That is, the following two statements are true for all sets $A$ and $B$, #$A,B\backslash in\backslash mathcal$ implies $A\; \backslash cap\; B\; \backslash in\; \backslash mathcal$ and #$A,B\backslash in\backslash mathcal$ implies $A\; \backslash cup\; B\; \backslash in\; \backslash mathcal.$〔.〕 In measure theory, a ring of sets $\backslash mathcal$ is instead a nonempty family closed under unions and set-theoretic differences.〔.〕 That is, the following two statements are true for all sets $A$ and $B$ (including when they are the same set), #$A,B\backslash in\backslash mathcal$ implies $A\; \backslash setminus\; B\; \backslash in\; \backslash mathcal$ and #$A,B\backslash in\backslash mathcal$ implies $A\; \backslash cup\; B\; \backslash in\; \backslash mathcal.$ This implies the empty set is in $\backslash mathcal$. It also implies that $\backslash mathcal$ is closed under symmetric difference and intersection, because of the identities #$A\backslash ,\backslash triangle\backslash ,B\; =\; (A\; \backslash setminus\; B)\; \backslash cup\; (B\; \backslash setminus\; A)$ and #$A\backslash cap\; B=A\backslash setminus(A\backslash setminus\; B).$ (So a ring in the second, measure theory, sense is also a ring in the first, order theory, sense.) Together, these operations give $\backslash mathcal$ the structure of a boolean ring. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and differences. This is due to the identities #$A\; \backslash cup\; B\; =\; (A\backslash ,\; \backslash triangle\backslash ,\; B)\backslash ,\; \backslash triangle\backslash ,\; (A\; \backslash cap\; B)$ and #$A\; \backslash setminus\; B\; =\; A\backslash ,\; \backslash triangle\backslash ,\; (A\; \backslash cap\; B).$ ==Examples== If ''X'' is any set, then the power set of ''X'' (the family of all subsets of ''X'') forms a ring of sets in either sense. If (''X'',≤) is a partially ordered set, then its upper sets (the subsets of ''X'' with the additional property that if ''x'' belongs to an upper set ''U'' and ''x'' ≤ ''y'', then ''y'' must also belong to ''U'') are closed under both intersections and unions. However, in general it will not be closed under differences of sets. The open sets and closed sets of any topological space are closed under both unions and intersections.〔 On the real line R, the family of sets consisting of the empty set and all finite unions of intervals of the form (''a'', ''b''], ''a'',''b'' in R is a ring in the measure theory sense. If ''T'' is any transformation defined on a space, then the sets that are mapped into themselves by ''T'' are closed under both unions and intersections.〔 If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form a ring of sets.〔 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「ring of sets」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.