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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 Dependence relation ： ウィキペディア英語版 Dependence relation In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence. Let $X$ be a set. A (binary) relation $\backslash triangleleft$ between an element $a$ of $X$ and a subset $S$ of $X$ is called a ''dependence relation'', written $a\; \backslash triangleleft\; S$, if it satisfies the following properties: * if $a\; \backslash in\; S$, then $a\; \backslash triangleleft\; S$; * if $a\; \backslash triangleleft\; S$, then there is a finite subset $S\_0$ of $S$, such that $a\; \backslash triangleleft\; S\_0$; * if $T$ is a subset of $X$ such that $b\; \backslash in\; S$ implies $b\; \backslash triangleleft\; T$, then $a\; \backslash triangleleft\; S$ implies $a\; \backslash triangleleft\; T$; * if $a\; \backslash triangleleft\; S$ but $a\; \backslash not\backslash !\backslash triangleleft\; S-\backslash lbrace\; b\; \backslash rbrace$ for some $b\; \backslash in\; S$, then $b\; \backslash triangleleft\; (S-\backslash lbrace\; b\; \backslash rbrace)\backslash cup\backslash lbrace\; a\; \backslash rbrace$. Given a ''dependence relation'' $\backslash triangleleft$ on $X$, a subset $S$ of $X$ is said to be ''independent'' if $a\; \backslash not\backslash !\backslash triangleleft\; S\; -\; \backslash lbrace\; a\; \backslash rbrace$ for all $a\; \backslash in\; S.$ If $S\; \backslash subseteq\; T$, then $S$ is said to ''span'' $T$ if $t\; \backslash triangleleft\; S$ for every $t\; \backslash in\; T.$ $S$ is said to be a ''basis'' of $X$ if $S$ is ''independent'' and $S$ ''spans'' $X.$ Remark. If $X$ is a non-empty set with a dependence relation $\backslash triangleleft$, then $X$ always has a basis with respect to $\backslash triangleleft.$ Furthermore, any two bases of $X$ have the same cardinality. ==Examples== * Let $V$ be a vector space over a field $F.$ The relation $\backslash triangleleft$, defined by $\backslash upsilon\; \backslash triangleleft\; S$ if $\backslash upsilon$ is in the subspace spanned by $S$, is a dependence relation. This is equivalent to the definition of linear dependence. * Let $K$ be a field extension of $F.$ Define $\backslash triangleleft$ by $\backslash alpha\; \backslash triangleleft\; S$ if $\backslash alpha$ is algebraic over $F(S).$ Then $\backslash triangleleft$ is a dependence relation. This is equivalent to the definition of algebraic dependence. 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「Dependence 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.