Partial function: Difference between revisions

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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'''.

MoreIn technicallyother words, a partial function is a [[binary relation]] over two [[Set (mathematics)|sets]] that associates to every element of the first set to ''at most'' one element of the second set; it is thus a [[Binary relation#Special types of binary relations|functional binaryunivalent relation]]. ItThis generalizes the concept of a (total) [[Function (mathematics)|function]] by not requiring ''every'' element of the first set to be associated to ''exactly'' onean element of the second set.

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.; Forin this reason, in calculus, and more generally in [[mathematical analysis]]context, a partial function is generally simply called simply a {{em|function}}.

In [[computability theory]], a [[general recursive function]] is a partial function from the integers to the integers; no [[algorithm]] can exist for deciding whether an arbitrary such function is in fact total.

When [[Function (mathematics)#Arrow notation|arrow notation]] is used for functions, a partial function <math>f</math> from <math>X</math> to <math>Y</math> is sometimes written as <math>f : X \rightharpoonup Y,</math> <math>f : X \nrightarrow Y,</math> or <math>f : X \hookrightarrow Y.</math> However, there is no general convention, and the latter notation is more commonly used for [[inclusion map]]s or [[embedding]]s.{{citation needed|reason=Provide a few example citations for each notation.|date=July 2019}}

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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}}

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[[Category:Mathematical relations]]

[[Category:Functions and mappings]]

[[Category:Properties of binary relations]]