Editing Function (mathematics) (section)

== Other terms ==
{{broader|Map (mathematics)}}

{| class="wikitable floatright" style= "width: 50%"
!Term
!Distinction from "function"
|-
| rowspan="3" |[[Map (mathematics)|Map/Mapping]]
|None; the terms are synonymous.<ref>{{Cite web|url=http://mathworld.wolfram.com/Map.html|title=Map|last=Weisstein|first=Eric W.|website=Wolfram MathWorld|language=en|access-date=2019-06-12}}</ref>
|-
|A map can have ''any set'' as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of [[real number|real]] or [[complex number|complex]] numbers.<ref name=Lang87p43>{{cite book |last=Lang |first=Serge |title=Linear Algebra |chapter=III §1. Mappings |chapter-url={{GBurl|0DUXym7QWfYC|p=43}} |publisher=Springer |date=1987 |isbn=978-0-387-96412-6 |edition=3rd |page=43 |quote=A function is a special type of mapping, namely it is a mapping from a set into the set of numbers, i.e. into, '''R''', or '''C''' or into a field ''K''.}}</ref>
|-
|Alternatively, a map is associated with a ''special structure'' (e.g. by explicitly specifying a structured codomain in its definition). For example, a [[linear map]].<ref name=Apostol81p35/>
|-
|[[Homomorphism]]
|A function between two [[structure (mathematics)|structures]] of the same type that preserves the operations of the structure (e.g. a [[group homomorphism]]).<ref>{{Cite book |last1=James |first1=Robert C. |author-link1=Robert C. James |title=Mathematics dictionary |last2=James |first2=Glenn |date=1992 |publisher=Van Nostrand Reinhold |isbn=0-442-00741-8 |edition=5th |page=202 |oclc=25409557}}</ref>
|-
|[[Morphism]]
|A generalisation of homomorphisms to any [[Category (mathematics)|category]], even when the objects of the category are not sets (for example, a [[group (mathematics)|group]] defines a category with only one object, which has the elements of the group as morphisms; see {{slink|Category (mathematics)|Examples}} for this example and other similar ones).<ref>{{harvnb|James|James|1992|p=48}}</ref>
|}

A function may also be called a '''map''' or a '''mapping''', but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. [[maps of manifolds]]). In particular ''map'' may be used in place of ''homomorphism'' for the sake of succinctness (e.g., [[linear map]] or ''map from {{mvar|G}} to {{mvar|H}}'' instead of ''[[group homomorphism]] from {{mvar|G}} to {{mvar|H}}''). Some authors<ref name=Apostol81p35>{{cite book |first=T. M. |last=Apostol |title=Mathematical Analysis|year=1981 |publisher=Addison-Wesley |page=35 |isbn=978-0-201-00288-1 |oclc=928947543 |edition=2nd}}</ref> reserve the word ''mapping'' for the case where the structure of the codomain belongs explicitly to the definition of the function.

Some authors, such as [[Serge Lang]],<ref name=Lang87p43/> use "function" only to refer to maps for which the [[codomain]] is a subset of the [[real number|real]] or [[complex number|complex]] numbers, and use the term ''mapping'' for more general functions.

In the theory of [[dynamical system]]s, a map denotes an [[Discrete-time dynamical system|evolution function]] used to create [[Dynamical system#Maps|discrete dynamical systems]]. See also [[Poincaré map]].

Whichever definition of ''map'' is used, related terms like ''[[Domain of a function|domain]]'', ''[[codomain]]'', ''[[Injective function|injective]]'', ''[[Continuous function|continuous]]'' have the same meaning as for a function.