翻訳と辞書 Words near each other ・ Equivalence of metrics ・ Equivalence partitioning ・ Equivalence point ・ Equivalence principle ・ Equivalence principle (geometric) ・ Equivalence relation ・ Equivalent (chemistry) ・ Equivalent air depth ・ Equivalent airspeed ・ Equivalent annual cost ・ Equivalent average ・ Equivalent carbon content ・ Equivalent circuit ・ Equivalent concentration ・ Equivalent Concrete Performance Concept ・ Equivalent definitions of mathematical structures ・ Equivalent dose ・ Equivalent dumping coefficient ・ Equivalent effective stratospheric chlorine ・ Equivalent impedance transforms ・ Equivalent input ・ Equivalent isotropically radiated power ・ Equivalent Lands ・ Equivalent latitude ・ Equivalent narcotic depth ・ Equivalent National Tertiary Entrance Rank ・ Equivalent noise resistance ・ Equivalent oxide thickness ・ Equivalent potential temperature ・ Equivalent pulse code modulation noise 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 Equivalent definitions of mathematical structures ： ウィキペディア英語版 Equivalent definitions of mathematical structures In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case). Second, a mathematical structure may have more than one definition (for example, topological space has at least 7 definitions; ordered field has at least 2 definitions). In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object. Many different objects may implement the same structure. ==Isomorphic implementations== Natural numbers may be implemented as 0 = , 1 = = , 2 = = }}, 3 = = , }}}} and so on; or alternatively as 0 = , 1 = =, 2 = = }} and so on. These are two different but isomorphic implementations of natural numbers in set theory. They are isomorphic as models of Peano axioms, that is, triples (''N'',0,''S'') where ''N'' is a set, 0 an element of ''N'', and ''S'' (called the successor function) a map of ''N'' to itself (satisfying appropriate conditions). In the first implementation ''S''(''n'') = ''n'' ∪ ; in the second implementation ''S''(''n'') = . As emphasized in Benacerraf's identification problem, the two implementations differ in their answer to the question whether 0 ∈ 2; however, this is not a legitimate question about natural numbers (since the relation ∈ is not stipulated by the relevant signature(s), see the next section).〔Technically, "0 ∈ 2" is an example of a non-transportable relation, see , .〕 Similarly, different but isomorphic implementations are used for complex numbers. 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「Equivalent definitions of mathematical structures」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.