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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 : : 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 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
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