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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 Codomain ： ウィキペディア英語版 Codomain In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The codomain is also sometimes referred to as the range but that term is ambiguous as it may also refer to the image. The codomain is part of a function if it is defined as described in 1954 by Nicolas Bourbaki, namely a triple , with a functional subset〔A set of pairs is ''functional'' iff no two distinct pairs have the same first component (''op. cit.'', p. 76 )〕 of the Cartesian product and is the set of first components of the pairs in (the ''domain''). The set is called the ''graph'' of the function. The set of all elements of the form , where ranges over the elements of the domain , is called the image of . In general, the image of a function is a subset of its codomain. Thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements in its codomain for which the equation does not have a solution. An alternative definition of ''function'' by Bourbaki (''op. cit.'', p. 77 ), namely as just a functional graph, does not include a codomain and is also widely used.〔, (pages 10–11 )〕 For example in set theory it is desirable to permit the domain of a function to be a proper class , in which case there is formally no such thing as a triple . With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form .〔, (quote 1 ), (quote 2 )〕〔, (page 8 )〕〔Mac Lane, in , (page 232 )〕〔, (page 91 )〕〔, (page 89 )〕 ==Examples== For a function :$f\backslash colon\; \backslash mathbb\backslash rightarrow\backslash mathbb$ defined by :$f\backslash colon\backslash ,x\backslash mapsto\; x^2$, or equivalently $f(x)\backslash \; =\backslash \; x^2$, the codomain of is $\backslash textstyle\; \backslash mathbb\; R$, but does not map to any negative number. Thus the image of is the set $\backslash textstyle\; \backslash mathbb^+\_0$; i.e., the interval . An alternative function is defined thus: : $g\backslash colon\backslash mathbb\backslash rightarrow\backslash mathbb^+\_0$ : $g\backslash colon\backslash ,x\backslash mapsto\; x^2.$ While and map a given to the same number, they are not, in this view, the same function because they have different codomains. A third function can be defined to demonstrate why: : $h\backslash colon\backslash ,x\backslash mapsto\; \backslash sqrt\; x.$ The domain of must be defined to be $\backslash textstyle\; \backslash mathbb^+\_0$: : $h\backslash colon\backslash mathbb^+\_0\backslash rightarrow\backslash mathbb$. The compositions are denoted :$h\; \backslash circ\; f$, :$h\; \backslash circ\; g$. On inspection, is not useful. It is true, unless defined otherwise, that the image of is not known; it is only known that it is a subset of $\backslash textstyle\; \backslash mathbb\; R$. For this reason, it is possible that , when composed with , might receive an argument for which no output is defined – negative numbers are not elements of the domain of , which is the square root function. Function composition therefore is a useful notion only when the ''codomain'' of the function on the right side of a composition (not its ''image'', which is a consequence of the function and could be unknown at the level of the composition) is the same as the domain of the function on the left side. The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, is a surjection while is not. The codomain does not affect whether a function is an injection. A second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from $\backslash textstyle\; \backslash mathbb^2$ to itself, which can be represented by the matrices with real coefficients. Each matrix represents a map with the domain $\backslash textstyle\; \backslash mathbb^2$ and codomain $\backslash textstyle\; \backslash mathbb^2$. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with rank ) but many do not, instead mapping into some smaller subspace (the matrices with rank or ). Take for example the matrix given by :$T\; =\; \backslash begin$ 1 & 0 \\ 1 & 0 \end which represents a linear transformation that maps the point to . The point is not in the image of , but is still in the codomain since linear transformations from $\backslash textstyle\; \backslash mathbb^2$ to $\backslash textstyle\; \backslash mathbb^2$ are of explicit relevance. Just like all matrices, represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that does not have full rank since its image is smaller than the whole codomain. 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「Codomain」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.