Map (mathematics)

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Map (mathematics)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and related technical fields, the term map or mapping is often a synonym

Synonym

Synonyms are different words with identical or very similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy....
for function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
. Thus, for example, a partial map is a partial function
Partial function

In mathematics, a partial function is a binary relation that associates each element of a Set , sometimes called its domain , with at most one element of another set, called its codomain....
, and a total map is a total function. Related terms like domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
, codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
, injective
Injective function

In mathematics, an injective function is a function which associates distinct arguments with distinct values.An injective function is called an injection, and is also said to be a one-to-one function ....
, continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
, etc. can be applied equally to maps and functions, with the same meaning.

In many branches of mathematics, the term is qualified with a property specific to that branch, such as a continuous function
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
in topology

Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, a linear map in linear algebra

Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, etc.

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Encyclopedia

In mathematics

Mathematics

Synonym

Topology

Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, etc. Correspondingly, in category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, 'map' is often used as a synonym for morphism or arrow.

Some authors such as Serge Lang

Serge Lang

Serge Lang was a France-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra....
use map as a general term for an association of an element in the range with every element in the domain, and function only to refer to maps in which the range is a field

Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
.

Sets of maps with special properties are the subjects of many important theories: see for instance Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
, mapping class group

Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space....
, permutation group

Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given Set M, and whose group operation is the composition of permutations in G ; the relationship is often written as ....
.

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

Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common....
in set theory

Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
.

In graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
, a map is a drawing of a graph

Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
on a surface without intersecting edges (a planar graph

Planar graph

In graph theory, a planar graph is a graph which can be graph embedding in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints....
).

In the theory of dynamical system

Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
s, a map denotes an evolution function used to create discrete dynamical systems

Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
. See also Poincaré map

Poincaré map

In mathematics, particularly in dynamical systems, a first recurrence map or Poincar? map, named after Henri Poincar?, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincar? section, Transversality to the Flow of the system....
.