Map (mathematics)

Function, homomorphism, or morphism

For other uses, see map (disambiguation).

A map is a function, as in the association of any of the four colored shapes in X to its color in Y

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.

## Maps as functions

[edit]

Main article: Function (mathematics)

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 have been given specific names. These include homomorphisms in 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

[edit]

Main article: Morphism

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.[9] 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 also

[edit]

* Apply 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 varieties

## References

[edit]

1. ^ 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.
2. ^ a b c d "Mapping | mathematics". Encyclopedia Britannica. Retrieved 2019-12-06.
3. ^ Apostol, T. M. (1981). Mathematical Analysis. Addison-Wesley. p. 35\. ISBN 0-201-00288-4.
4. ^ Stacho, Juraj (October 31, 2007). "Function, one-to-one, onto" (PDF). cs.toronto.edu. Retrieved 2019-12-06.
5. ^ "Functions or Mapping | Learning Mapping | Function as a Special Kind of Relation". Math Only Math. Retrieved 2019-12-06.
6. ^ Weisstein, Eric W. "Map". mathworld.wolfram.com. Retrieved 2019-12-06.
7. ^ "Mapping, Mathematical | Encyclopedia.com". www.encyclopedia.com. Retrieved 2019-12-06.
8. ^ Lang, Serge (1971). Linear Algebra (2nd ed.). Addison-Wesley. p. 83\. ISBN 0-201-04211-8.
9. ^ Simmons, H. (2011). An Introduction to Category Theory. Cambridge University Press. p. 2\. ISBN 978-1-139-50332-7.

### Works cited

[edit]

* Halmos, Paul R. (1970). Naive Set Theory. Springer-Verlag. ISBN 978-0-387-90092-6.

## External links

[edit]

* v
* t
* e

Mathematical logic

General|

* Axiom
* list
* Cardinality
* First-order logic
* Formal proof
* Formal semantics
* Foundations of mathematics
* Information theory
* Lemma
* Logical consequence
* Model
* Theorem
* Theory
* Type theory

Theorems (list)
and paradoxes|

* Gödel's completeness and incompleteness theorems
* Tarski's undefinability
* Banach–Tarski paradox
* Cantor's theorem, paradox and diagonal argument
* Compactness
* Halting problem
* Lindström's
* Löwenheim–Skolem
* Russell's paradox

Logics|

| Traditional|

* Classical logic
* Logical truth
* Tautology
* Proposition
* Inference
* Logical equivalence
* Consistency
* Equiconsistency
* Argument
* Soundness
* Validity
* Syllogism
* Square of opposition
* Venn diagram

|

Propositional|

* Boolean algebra
* Boolean functions
* Logical connectives
* Propositional calculus
* Propositional formula
* Truth tables
* Many-valued logic
* 3
* finite
* ∞

Predicate|

* First-order
* list
* Second-order
* Monadic
* Higher-order
* Fixed-point
* Free
* Quantifiers
* Predicate
* Monadic predicate calculus

Set theory|

* Set
* hereditary
* Class
* (Ur-)Element
* Ordinal number
* Extensionality
* Forcing
* Relation
* equivalence
* partition
* Set operations:
* intersection
* union
* complement
* Cartesian product
* power set
* identities

Types of sets|

* Countable
* Uncountable
* Empty
* Inhabited
* Singleton
* Finite
* Infinite
* Transitive
* Ultrafilter
* Recursive
* Fuzzy
* Universal
* Universe
* constructible
* Grothendieck
* Von Neumann

Maps and cardinality|

* Function/Map
* domain
* codomain
* image
* In/Sur/Bi-jection
* Schröder–Bernstein theorem
* Isomorphism
* Gödel numbering
* Enumeration
* Large cardinal
* inaccessible
* Aleph number
* Operation
* binary

Set theories|

* Zermelo–Fraenkel
* axiom of choice
* continuum hypothesis
* General
* Kripke–Platek
* Morse–Kelley
* Naive
* New Foundations
* Tarski–Grothendieck
* Von Neumann–Bernays–Gödel
* Ackermann
* Constructive

Formal systems (list),
language and syntax|

* Alphabet
* Arity
* Automata
* Axiom schema
* Expression
* ground
* Extension
* by definition
* conservative
* Relation
* Formation rule
* Grammar
* Formula
* atomic
* closed
* ground
* open
* Free/bound variable
* Language
* Metalanguage
* Logical connective
* ¬
* ∨
* ∧
* →
* ↔
* =
* Predicate
* functional
* variable
* propositional variable
* Proof
* Quantifier
* ∃
* !
* ∀
* rank
* Sentence
* atomic
* spectrum
* Signature
* String
* Substitution
* Symbol
* function
* logical/constant
* non-logical
* variable
* Term
* Theory
* list

Example axiomatic
systems (list)|

* of arithmetic:
* Peano
* second-order
* elementary function
* primitive recursive
* Robinson
* Skolem
* of the real numbers
* Tarski's axiomatization
* of Boolean algebras
* canonical
* minimal axioms
* of geometry:
* Euclidean:
* Elements
* Hilbert's
* Tarski's
* non-Euclidean

* Principia Mathematica

Proof theory|

* Formal proof
* Natural deduction
* Logical consequence
* Rule of inference
* Sequent calculus
* Theorem
* Systems
* axiomatic
* deductive
* Hilbert
* list
* Complete theory
* Independence (from ZFC)
* Proof of impossibility
* Ordinal analysis
* Reverse mathematics
* Self-verifying theories

Model theory|

* Interpretation
* function
* of models
* Model
* equivalence
* finite
* saturated
* spectrum
* submodel
* Non-standard model
* of arithmetic
* Diagram
* elementary
* Categorical theory
* Model complete theory
* Satisfiability
* Semantics of logic
* Strength
* Theories of truth
* semantic
* Tarski's
* Kripke's
* T-schema
* Transfer principle
* Truth predicate
* Truth value
* Type
* Ultraproduct
* Validity

Computability theory|

* Church encoding
* Church–Turing thesis
* Computably enumerable
* Computable function
* Computable set
* Decision problem
* decidable
* undecidable
* P
* NP
* P versus NP problem
* Kolmogorov complexity
* Lambda calculus
* Primitive recursive function
* Recursion
* Recursive set
* Turing machine
* Type theory

Related|

* Abstract logic
* Algebraic logic
* Automated theorem proving
* Category theory
* Concrete/Abstract category
* Category of sets
* History of logic
* History of mathematical logic
* timeline
* Logicism
* Mathematical object
* Philosophy of mathematics
* Supertask

Mathematics portal

Authority control databases

National|

* United States
* Czech Republic
* Latvia
* Israel

Other|

* Yale LUX

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template