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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 Partial equivalence relation ： ウィキペディア英語版 Partial equivalence relation In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) $R$ on a set $X$ is a relation that is ''symmetric'' and ''transitive''. In other words, it holds for all $a,\; b,\; c\; \backslash in\; X$ that: # if $a\; R\; b$, then $b\; R\; a$ (symmetry) # if $a\; R\; b$ and $b\; R\; c$, then $a\; R\; c$ (transitivity) If $R$ is also reflexive, then $R$ is an equivalence relation. == Properties and applications == In a set-theoretic context, there is a simple structure to the general PER $R$ on $X$: it is an equivalence relation on the subset $Y\; =\; \backslash \; \backslash subseteq\; X$. ($Y$ is the subset of $X$ such that in the complement of $Y$ ($X\backslash setminus\; Y$) no element is related by $R$ to any other.) By construction, $R$ is reflexive on $Y$ and therefore an equivalence relation on $Y$. Notice that $R$ is actually only true on elements of $Y$: if $x\; R\; y$, then $y\; R\; x$ by symmetry, so $x\; R\; x$ and $y\; R\; y$ by transitivity. Conversely, given a subset ''Y'' of ''X'', any equivalence relation on ''Y'' is automatically a PER on ''X''. PERs are therefore used mainly in computer science, type theory and constructive mathematics, particularly to define setoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to the operations of subset and quotient in classical set-theoretic mathematics. Every partial equivalence relation is a difunctional relation, but the converse does not hold. The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive. 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「Partial equivalence 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.