Map (mathematics)

For other uses, see Map (disambiguation).

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function. There are also a few, less common uses in logic and graph theory.

Contents

• 
  1 Maps as functions
  


• 
  2 Maps as morphisms
  


• 
  3 Other uses
  
  
  
  • 
    3.1 In logic
    
  
  
  • 
    3.2 In graph theory
    
  
  
  • 
    3.3 In computer science
    
  
  
  
  


• 
  4 See also
  


• 
  5 References
  


• 
  6 External links
  

Maps as functions

Main article: Function (mathematics)

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,^[1] use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of the fields R or C) and the term mapping for more general functions.

Sets of maps of special kinds are the subjects of many important theories: see for instance Lie group, mapping class group, permutation group.

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.

A partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. 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.

Maps as morphisms

Main article: Morphism

In category theory, "map" is often used as a synonym for morphism or arrow, thus for something more general than a function.^[2] For example, morphisms fX→Y{displaystyle f,Xto Y}, in a concrete category, in other words morphisms that can be viewed as functions, carry with them the information of both its domain (the source X{displaystyle X} of the morphism), but also its co-domain (the target Y{displaystyle Y}). In the widely used definition of function f:X→Y{displaystyle f:Xto Y}, this is a subset of X×Y{displaystyle Xtimes Y} consisting of all the pairs (x,f(x)){displaystyle (x,f(x))} for x∈X{displaystyle xin X}. In this sense, the function doesn't capture the information of which set Y{displaystyle Y} is used as the co-domain. Only the range f(X){displaystyle f(X)} is determined by the function.

Other uses

In logic

In formal logic, the term map is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.

In graph theory

An example of a map in graph theory

In graph theory, a map is a drawing of a graph on a surface without overlapping edges (an embedding). If the surface is a plane then a map is a planar graph, similar to a political map.^[3]

In computer science

In the communities surrounding programming languages that treat functions as first-class citizens, a map often refers to the binary higher-order function that takes a function f and a list [v₀, v₁, ..., v_n] as arguments and returns [f(v₀), f(v₁), ..., f(v_n)], where n ≥ 0.

See also

• Bijection, injection and surjection


• Category theory


• Correspondence (mathematics)


• Homeomorphism


• Homomorphism


• List of chaotic maps


• Mapping class group


• Morphism


• Projection (mathematics)


• Topology

References

1. 
   ^ Lang, Serge (1971), Linear Algebra (2nd ed.), Addison-Wesley, p. 83.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
   


2. 
   ^ Simmons, H. (2011), An Introduction to Category Theory, Cambridge University Press, p. 2, ISBN 9781139503327
   


3. 
   ^ Gross, Jonathan; Yellen, Jay (1998), Graph Theory and its applications, CRC Press, p. 294, ISBN 0-8493-3982-0
   

External links

• 
  Media related to Mappings - functions at Wikimedia Commons

v t e Mathematical logic
General	Formal language Formation rule Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Rule of inference Relation Theorem Logical consequence Type theory Symbol Syntax Theory
Systems	Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus
Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram
Propositional calculus and Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic
Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus
Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair
Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory
Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity
Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax
Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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