In

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, a map is often used as a synonym for a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

, but may also refer to some generalizations. Originally, this was an abbreviation of mapping, which often refers to the action of applying a function to the elements of its domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

. This terminology is not completely fixed, as these terms are generally not formally defined, and can be considered to be jargon
Jargon is the specialized terminology
Terminology is a general word for the group of specialized words or meanings relating to a particular field, and also the study of such terms and their use. This is also known as terminology science. Terms a ...

. These terms may have originated as a generalization of the process of making a geographical map
Cartography (; from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10 ...

, which consists of ''mapping'' the Earth surface to a sheet of paper.
Maps may either be ''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

s'' or ''morphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s'', though the terms share some overlap. The term ''map'' may be used to distinguish some special types of functions, such as homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homom ...

s. For example, a linear map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is a homomorphism of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, while the term linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomial ...

may have this meaning as well as another one. In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...

, a map may refer to a morphism, which is a generalization of the idea of a function. In some occasions, the term ''transformation'' can also be used interchangeably. There are also a few less common uses in logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit ...

and graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' ( ...

.
Maps as functions

In many branches of mathematics, the term ''map'' is used to mean afunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...

" in topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

, a "linear transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

" in linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (math ...

, etc.
Some authors, such as Serge Lang
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

, use "function" only to refer to maps in which the codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a set of numbers (i.e. a subset of 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
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, isometries
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

in geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...

, operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (called ''operands'') to a well-defined output value. The ...

in analysis
Analysis is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Ari ...

and representations
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It co ...

in group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...

.
In the theory of dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the s ...

s, a map denotes an evolution function
The dynamical system concept is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Sou ...

used to create .
A ''partial map'' is a ''partial function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

''. Related terms such as ''domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

'', ''codomain'', ''injective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

'', and ''continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

'' 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

" or "arrow", and thus is more general than "function". For example, a morphism $f:\backslash ,\; X\; \backslash to\; Y$ in a concrete category
In mathematics, a concrete category is a category (category theory), category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see #Relative concreteness, Relative concreteness below''). This func ...

(i.e. a morphism which can be viewed as functions) carries with it the information of its domain (the source $X$ of the morphism) and its codomain (the target $Y$). In the widely used definition of a function $f:X\backslash to\; Y$, $f$ is a subset of $X\backslash times\; Y$ consisting of all the pairs $(x,f(x))$ for $x\backslash in\; X$. In this sense, the function does not capture the information of which set $Y$ is used as the codomain; only the range $f(X)$ is determined by the function.
Other uses

In logic

Informal logic
Logic is an interdisciplinary field which studies truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to r ...

, the term ''map'' is sometimes used for a ''functional predicate
In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term.
Functional predicates are also sometimes called mappings, but ...

'', whereas a function is a model
In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...

of such a predicate
Predicate or predication may refer to:
Computer science
*Syntactic predicate (in parser technology) guidelines the parser process
Linguistics
*Predicate (grammar), a grammatical component of a sentence
Philosophy and logic
* Predication (philo ...

in set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...

.
In graph theory

Ingraph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' ( ...

, a ''map'' is a drawing of a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

on a surface without overlapping edges (an embedding
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

). If the surface is a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse
*Plane (Magic: Th ...

then a map is a planar graph
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph t ...

, similar to a political map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as Physical body, objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interac ...

.
In computer science

In the communities surroundingprogramming language
A programming language is a formal language comprising a Instruction set architecture, set of instructions that produce various kinds of Input/output, output. Programming languages are used in computer programming to implement algorithms.
Most ...

s that treat functions as first-class citizen
In programming language design, a first-class citizen (also type, object, entity, or value) in a given programming language
A programming language is a formal language comprising a Instruction set architecture, set of instructions that produce ...

s, a map
A map is a symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, Object (philosophy), object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or s ...

is often referred to as the binary
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics, a binary number is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal ...

higher-order function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

that takes a function ''f'' and a list
A ''list'' is any set of items. List or lists may also refer to:
People
* List (surname)List or Liste is a European surname. Notable people with the surname include:
List
* Friedrich List (1789–1846), German economist
* Garrett List (1943 ...

as arguments
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...

and returns (where ).
See also

* * * * * *List of chaotic maps
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

*
References

External links

{{authority controlBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...