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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 Ordered pair ： ウィキペディア英語版 Ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of mathematical objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In contrast, the unordered pair equals the unordered pair .) Ordered pairs are also called 2-tuples, or sequences of length 2; ordered pairs of scalars are also called 2-dimensional vectors. The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ''second entry'' of the pair. Alternatively, the objects are called the first and second ''coordinates'', or the left and right ''projections'' of the ordered pair. Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs. ==Generalities== Let $(a\_1,\; b\_1)$ and $(a\_2,\; b\_2)$ be ordered pairs. Then the characteristic (or ''defining'') property of the ordered pair is: :$(a\_1,\; b\_1)\; =\; (a\_2,\; b\_2)\backslash quad\backslash text\backslash quad\; a\_1\; =\; a\_2\backslash textb\_1\; =\; b\_2.\backslash !$ The set of all ordered pairs whose first entry is in some set ''A'' and whose second entry is in some set ''B'' is called the Cartesian product of ''A'' and ''B'', and written ''A'' × ''B''. A binary relation between sets ''A'' and ''B'' is a subset of ''A'' × ''B''. If one wishes to employ the $\backslash \; (a,b)$ notation for a different purpose (such as denoting open intervals on the real number line) the ordered pair may be denoted by the variant notation $\backslash left\; \backslash langle\; a,b\backslash right\; \backslash rangle.$ The left and right projection of a pair ''p'' is usually denoted by ''π''1(''p'') and ''π''2(''p''), or by ''π''''l''(''p'') and ''π''''r''(''p''), respectively. In contexts where arbitrary ''n''-tuples are considered, ''π''''n''''i''(''t'') is a common notation for the ''i''-th component of an ''n'' tuple ''t''. 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「Ordered pair」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.