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Undid revision 1250997369 by Closed Limelike Curves (talk) this is true of only one of the two bolded phrases, but also (1) it does not require it in the first paragraph, and (2) if in fact if the term is not mentioned anywhere in this article then a more plausible conclusion is that the redirect should not exist |
Undid revision 1291265531 by Goodphy (talk): contradicts the terminlogy just introduced ("___domain" means S) |
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In [[mathematics]], a '''partial function''' {{mvar|f}} from a [[Set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} is a [[function (mathematics)|function]] from a [[subset]] {{mvar|S}} of {{mvar|X}} (possibly the whole {{mvar|X}} itself) to {{mvar|Y}}. The subset {{mvar|S}}, that is, the ''[[Domain of a function|___domain]]'' of {{mvar|f}} viewed as a function, is called the '''___domain of definition''' or '''natural ___domain''' of {{mvar|f}}. If {{mvar|S}} equals {{mvar|X}}, that is, if {{mvar|f}} is defined on every element in {{mvar|X}}, then {{mvar|f}} is said to be a '''total function'''.
A partial function is often used when its exact ___domain of definition is not known, or is difficult to specify. However, even when the exact ___domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in [[calculus]], where, for example, the [[quotient]] of two functions is a partial function whose ___domain of definition cannot contain the [[Zero of a function|zeros]] of the denominator; in this context, a partial function is generally simply called a {{em|function}}.
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=== In category theory ===
In [[category theory]], when considering the operation of [[morphism]] composition in [[concrete categories]], the composition operation <math>\circ \;:\; \hom(C) \times \hom(C) \to \hom(C)</math> is a total function if and only if <math>\operatorname{ob}(C)</math> has one element. The reason for this is that two morphisms <math>f : X \to Y</math> and <math>g : U \to V</math> can only be composed as <math>g \circ f</math> if <math>Y = U,</math> that is, the codomain of <math>f</math> must equal the ___domain of <math>g.</math>
The category of sets and partial functions is [[Equivalence of categories|equivalent]] to but not [[Isomorphism of categories|isomorphic]] with the category of [[pointed set]]s and point-preserving maps.<ref name="KoslowskiMelton2001">{{cite book|editor=Jürgen Koslowski and Austin Melton|title=Categorical Perspectives|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4186-3|page=10|author=Lutz Schröder|chapter=Categories: a free tour}}</ref> One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology ([[one-point compactification]]) and in [[theoretical computer science]]."<ref name="KoblitzZilber2009">{{cite book|author1=Neal Koblitz|author2=B. Zilber|author3=Yu. I. Manin|title=A Course in Mathematical Logic for Mathematicians|year=2009|publisher=Springer Science & Business Media|isbn=978-1-4419-0615-1|page=290}}</ref>
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== See also ==
{{Functions}}▼
* {{annotated link|Analytic continuation}}
* {{annotated link|Multivalued function}}
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== References ==
{{reflist|group=note}}▼
{{reflist}}▼
* [[Martin Davis (mathematician)|Martin Davis]] (1958), ''Computability and Unsolvability'', McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. {{isbn|0-486-61471-9}}.
* [[Stephen Kleene]] (1952), ''Introduction to Meta-Mathematics'', North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). {{isbn|0-7204-2103-9}}.
* [[Harold S. Stone]] (1972), ''Introduction to Computer Organization and Data Structures'', McGraw–Hill Book Company, New York.
=== Notes ===
▲{{reflist|group=note}}
▲{{reflist}}
▲{{Functions navbox}}
[[Category:Mathematical relations]]
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