In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.
The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory.
Maps as functions
In many branches of mathematics, the term map is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc.
Some authors, such as Serge Lang, use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of R or C), and reserve the term mapping for more general functions.
Maps of certain kinds have been given specific names. These include homomorphisms in algebra, isometries in geometry, operators in analysis and representations in group theory.
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems.
A partial map is a partial function. Related terminology such as domain, codomain, injective, and continuous can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.
As morphisms
In category theory, "map" is often used as a synonym for "morphism" or "arrow", which is a structure-respecting function and thus may imply more structure than "function" does. For example, a morphism 
See also
- Apply function – Function that maps a function and its arguments to the function value
- Arrow notation – e.g.,
, also known as map
- Bijection, injection and surjection – Properties of mathematical functions
- Homeomorphism – Mapping which preserves all topological properties of a given space
- List of chaotic maps
- Maplet arrow (↦) – commonly pronounced "maps to"
- Mapping class group – Group of isotopy classes of a topological automorphism group
- Permutation group – Group whose operation is composition of permutations
- Regular map (algebraic geometry) – Morphism of algebraic varieties
References
- The words map, mapping, correspondence, and operator are often used synonymously. Halmos 1970, p. 30. Some authors use the term function with a more restricted meaning, namely as a map that is restricted to apply to numbers only.
- "Mapping | mathematics". Encyclopedia Britannica. Retrieved 2019-12-06.
- Apostol, T. M. (1981). Mathematical Analysis. Addison-Wesley. p. 35. ISBN 0-201-00288-4.
- Stacho, Juraj (October 31, 2007). "Function, one-to-one, onto" (PDF). cs.toronto.edu. Retrieved 2019-12-06.
- "Functions or Mapping | Learning Mapping | Function as a Special Kind of Relation". Math Only Math. Retrieved 2019-12-06.
- Weisstein, Eric W. "Map". mathworld.wolfram.com. Retrieved 2019-12-06.
- "Mapping, Mathematical | Encyclopedia.com". www.encyclopedia.com. Retrieved 2019-12-06.
- Lang, Serge (1971). Linear Algebra (2nd ed.). Addison-Wesley. p. 83. ISBN 0-201-04211-8.
- Simmons, H. (2011). An Introduction to Category Theory. Cambridge University Press. p. 2. ISBN 978-1-139-50332-7.
Works cited
- Halmos, Paul R. (1970). Naive Set Theory. Springer-Verlag. ISBN 978-0-387-90092-6.
External links
In mathematics a map or mapping is a function in its general sense These terms may have originated as from the process of making a geographical map mapping the Earth surface to a sheet of paper A map is a function as in the association of any of the four colored shapes in X to its color in Y The term map may be used to distinguish some special types of functions such as homomorphisms For example a linear map is a homomorphism of vector spaces while the term linear function may have this meaning or it may mean a linear polynomial In category theory a map may refer to a morphism The term transformation can be used interchangeably but transformation often refers to a function from a set to itself There are also a few less common uses in logic and graph theory Maps as functionsIn many branches of mathematics the term map is used to mean a function sometimes with a specific property of particular importance to that branch For instance a map is a continuous function in topology a linear transformation in linear algebra etc Some authors such as Serge Lang use function only to refer to maps in which the codomain is a set of numbers i e a subset of R or C and reserve the term mapping for more general functions Maps of certain kinds have been given specific names These include homomorphisms in algebra isometries in geometry operators in analysis and representations in group theory In the theory of dynamical systems a map denotes an evolution function used to create discrete dynamical systems A partial map is a partial function Related terminology such as domain codomain injective and continuous can be applied equally to maps and functions with the same meaning All these usages can be applied to maps as general functions or as functions with special properties As morphismsIn category theory map is often used as a synonym for morphism or arrow which is a structure respecting function and thus may imply more structure than function does For example a morphism f X Y displaystyle f X to Y in a concrete category i e a morphism that can be viewed as a function carries with it the information of its domain the source X displaystyle X of the morphism and its codomain the target Y displaystyle Y In the widely used definition of a function f X Y displaystyle f X to Y f displaystyle f is a subset of X Y displaystyle X times Y consisting of all the pairs x f x displaystyle x f x for x X displaystyle x in X In this sense the function does not capture the set Y displaystyle Y that is used as the codomain only the range f X displaystyle f X is determined by the function See alsoApply function Function that maps a function and its arguments to the function value Arrow notation e g x x 1 displaystyle x mapsto x 1 also known as map Bijection injection and surjection Properties of mathematical functions Homeomorphism Mapping which preserves all topological properties of a given space List of chaotic maps Maplet arrow commonly pronounced maps to Mapping class group Group of isotopy classes of a topological automorphism group Permutation group Group whose operation is composition of permutations Regular map algebraic geometry Morphism of algebraic varietiesReferencesThe words map mapping correspondence and operator are often used synonymously Halmos 1970 p 30 Some authors use the term function with a more restricted meaning namely as a map that is restricted to apply to numbers only Mapping mathematics Encyclopedia Britannica Retrieved 2019 12 06 Apostol T M 1981 Mathematical Analysis Addison Wesley p 35 ISBN 0 201 00288 4 Stacho Juraj October 31 2007 Function one to one onto PDF cs toronto edu Retrieved 2019 12 06 Functions or Mapping Learning Mapping Function as a Special Kind of Relation Math Only Math Retrieved 2019 12 06 Weisstein Eric W Map mathworld wolfram com Retrieved 2019 12 06 Mapping Mathematical Encyclopedia com www encyclopedia com Retrieved 2019 12 06 Lang Serge 1971 Linear Algebra 2nd ed Addison Wesley p 83 ISBN 0 201 04211 8 Simmons H 2011 An Introduction to Category Theory Cambridge University Press p 2 ISBN 978 1 139 50332 7 Works cited Halmos Paul R 1970 Naive Set Theory Springer Verlag ISBN 978 0 387 90092 6 External links
