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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 function ： ウィキペディア英語版 Partial function In mathematics, a partial function from ''X'' to ''Y'' (written as ) is a function , for some subset ''X''′ of ''X''. It generalizes the concept of a function by not forcing ''f'' to map ''every'' element of ''X'' to an element of ''Y'' (only some subset ''X''′ of ''X''). If , then ''f'' is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, ''X''′, is not known (e.g. many functions in computability theory). Specifically, we will say that for any , either: * (it is defined as a single element in ''Y'') or * ''f''(''x'') is undefined. For example we can consider the square root function restricted to the integers :$g\backslash colon\; \backslash mathbb\; \backslash to\; \backslash mathbb$ :$g(n)\; =\; \backslash sqrt.$ Thus ''g''(''n'') is only defined for ''n'' that are perfect squares (). So, , but ''g''(26) is undefined. == Basic concepts == There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the term "domain of ''f''" for the set of all values ''x'' such that ''f''(''x'') is defined (''X''' above). But some, particularly category theorists, consider the domain of a partial function ''f'':''X'' → ''Y'' to be ''X'', and refer to ''X''' as the domain of definition. Similarly, the term range can refer to either the codomain or the ''image'' of a function. Occasionally, a partial function with domain ''X'' and codomain ''Y'' is written as ''f'': ''X'' ⇸ ''Y'', using an arrow with vertical stroke. A partial function is said to be injective or surjective when the total function given by the restriction of the partial function to its domain of definition is. A partial function may be both injective and surjective. Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse. Furthermore, a total function which is injective may be inverted to an injective partial function. The notion of transformation can be generalized to partial functions as well. A partial transformation is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' are subsets of some set ''X''. 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 function」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.