In mathematics, a map or mapping is a function in its general sense.^{[1]} These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.^{[2]}
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.^{[3]}^{[4]} In category theory, a map may refer to a morphism.^{[2]} The term transformation can be used interchangeably,^{[2]} 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.
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14.1 Definition of a Mapping (Basic Mathematics)

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Maps as functions
In many branches of mathematics, the term map is used to mean a function,^{[5]}^{[6]}^{[7]} 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,^{[8]} 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 are the subjects of many important theories. These include homomorphisms in abstract algebra, isometries in geometry, operators in analysis and representations in group theory.^{[2]}
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 structurerespecting function and thus may imply more structure than "function" does.^{[9]} For example, a morphism 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 of the morphism) and its codomain (the target ). In the widely used definition of a function , is a subset of consisting of all the pairs for . In this sense, the function does not capture the set that is used as the codomain; only the range is determined by the function.
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.
 ^ ^{a} ^{b} ^{c} ^{d} "Mapping  mathematics". Encyclopedia Britannica. Retrieved 20191206.
 ^ Apostol, T. M. (1981). Mathematical Analysis. AddisonWesley. p. 35. ISBN 0201002884.
 ^ Stacho, Juraj (October 31, 2007). "Function, onetoone, onto" (PDF). cs.toronto.edu. Retrieved 20191206.
 ^ "Functions or Mapping  Learning Mapping  Function as a Special Kind of Relation". Math Only Math. Retrieved 20191206.
 ^ Weisstein, Eric W. "Map". mathworld.wolfram.com. Retrieved 20191206.
 ^ "Mapping, Mathematical  Encyclopedia.com". www.encyclopedia.com. Retrieved 20191206.
 ^ Lang, Serge (1971). Linear Algebra (2nd ed.). AddisonWesley. p. 83. ISBN 0201042118.
 ^ Simmons, H. (2011). An Introduction to Category Theory. Cambridge University Press. p. 2. ISBN 9781139503327.
Works cited
 Halmos, Paul R. (1970). Naive Set Theory. SpringerVerlag. ISBN 9780387900926.