Map

Definition: Map

Map

Noun

1. A diagrammatic representation of the earth's surface (or part of it).

2. A function such that for every element of one set there is a unique element of another set.

Verb

1. Make a map of; show or establish the features of details of: "map the surface of Venus".

2. Explore or survey for the purpose of making a map; "We haven't even begun to map the many galaxies that we know exist".

3. Locate within a specific region of a chromosome in relation to known DNA or gene sequences; "map the genes".

4. Plan, delineate, or arrange in detail; ""map one's future".

5. Depict as if on a map; "sorrow was mapped on the mother's face".

6. To establish a mapping (of mathematical elements or sets).

Date "map" was first used in popular English literature: sometime before 1374. (references)

Specialty Definition: Map

Domain	Definition
Computing	MAP 1. Manufacturing Automation Protocol. 2. Mathematical Analysis without Programming. (1996-12-01) map 1. function. 2. In functional programming, the most common higher-order function over lists. Map applies its first argument to each element of its second argument (a list) and returns the list of results. map :: (a -> b) -> [a] -> [b] map f [] = [] map f (x:xs) = f x : map f xs This can be generalised to types other than lists. (1997-11-05). Source: The Free On-line Dictionary of Computing.
Agriculture	Market Access Program. (references)
Dream Interpretation	To dream of a map, or studying one, denotes a change will be contemplated in your business. Some disappointing things will occur, but much profit also will follow the change. To dream of looking for one, denotes that a sudden discontent with your surroundings will inspire you with new energy, and thus you will rise into better conditions. For a young woman, this dream denotes that she will rise into higher spheres by sheer ambition. Source: Ten Thousand Dreams Interpreted ....
Geography	A graphic representation, usually on a plane surface, and an established scale, of natural or artificial features on the surface of a part or the whole of the earth or other planetary body. The features are positioned relative to a coordinate reference system. Source: European Union. (references)
Hydrologic	The average rainfall over a given area, generally expressed as an average depth over the area. MAP (Mean Areal Precipitation). (references)
Mining	A. A horizontal projection of surface plants, mine workings, or both, drawn to a definite scale, upon which is shown all the important features of the mine; a plan; a plat. b. The act of preparing such plans of a mine. c. A representation to a definite scale on a horizontal plane of the physical features of a portion of the Earth's surface (natural or artificial or both) by means of symbols, which may emphasize, generalize, or omit certain features as conditions may warrant. A map may be derived from ground surveys made by transit, plane table, or camera, or fromaerial photographic surveys, or both. (references)
Source: compiled by the editor from various references; see credits.

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Specialty Definition: Cartography

(From Wikipedia, the free Encyclopedia)

simple:Cartography
Cartography (or mapmaking) is the study and practice of making maps or globes. Maps have traditionally been made using pen and paper, but the advent and spread of computers has revolutionized cartography. Most commercial quality maps are now made with map making software that falls into one of three main types; CAD, GIS, and specialized map illustration software.
Maps function as visualization tools for spatial data. Spatial data is acquired from measurement and can be stored in a database, from which it can be extracted for a variety of purposes. Current trends in this field are moving away from analog methods of mapmaking and toward the creation of increasingly dynamic, interoperable maps that can be manipulated digitally. The cartographic process rests on the premise that there is there is an objective reality and that we can make reliable representations of that reality by adding levels of abstraction.

History
The oldest known map dates from the 5000s BC. These more primitive maps emphasize topological relationships such as connectedness, adjacency and containment.
A major development in mapmaking occurred with the advent of geometry which was first used in Babylonia around 2300s BC. An engraved map of the holy city of Nippur, from the Kassite period (14th - 12 centuries BCE) of Babylonian history, was found at Nippur [1]. The Egyptiansians later used geometry to survey land and to resurvey it after the periodic flooding of the Nile obscured the property borders.
The ancient Greeks added a great deal to the art and science of cartography. Strabo (c. 63 BCE - c CE 21) is credited as the father of geography because he wrote "Geographia" in which he documented and criticized the works of others (most of whom would not be known today had Strabo not mentioned them). Thales of Miletus thought that the earth was disk and was supported by water in around 600 BC. Anaximander of Miletus theorized that the earth was cylindrical also about the same time. In 288 BCE Aristarchus of Samos was the first to say that the sun was the center of universe (see heliocentric theory). And in approximately 250 BC Eratosthenes of Cyrene estimated the circumference of the earth to within 15% of the modern-day accepted value.
Pythagoras of Ionia, who was the founder of a mathematical cult that developed many number-based superstitions that later became the basis of mathematics, was the first notable person to say that the earth was a sphere. Aristotle later provided arguments in support of this idea. Those arguments can be summarized as follows:
The lunar eclipse is always circular.
Ships seem to sink as they move away from view and pass the horizon.
Some stars can only be seen from certain parts of the earth.

The Greeks also developed the science of map projections, which are methods of representing the curved surface of the earth on a plane. Eratosthenes, Anaximander, and Hipparchus are credited with developing the concept of longitude and latitude, and Eratosthenes seems to have developed the equirectangular map projection around 200 BC. Claudius Ptolemy developed map projections as well, including the equidistant conic around 150 BC.
During the middle agess of Europe intellectual thought tended toward religion. While scientific cartography advanced in some ways, such as Roger Bacon's investigations of map projections and the appearance of portolano and then portolan charts for plying the European trade routes, there was little impetus for systematic study of cartography. Most world 'maps' of the period were Christian cosmological diagrams that were not intended to be rigorous geographical representations. They were typically rectangular or circular and followed the style of the so-called "T and O map." This world map represented the land as disk-shaped and surrounded by Ocean. The land on the map was divided into three parts by a T shape in which Asia occupied the top of the T area, Europe the bottom left and Africa the bottom right. Dogma also dictated that one son of Abraham colonized each division. The Chinese during this time were using a rectangular coordinate system which was far more accurate and useful.
The discovery of the West by Europeans and the subsequent effort to control and divide up those lands necessitated the invention of scientific mapping methods. The trend of globalism that was started with the Age of Exploration would continue during the Renaissance. This would, in turn, would eventually lead to the Enlightenment in which probability theory, a concern for accuracy, and a desire to classify the world would further develop scientific mapmaking. The concept of distribution, in which systems are characterized and analyzed, and ecological thinking, in which the interrelationships between objects are studied and predictions are made about future behavior, would revolutionize cartography in later centuries.

Technological Changes
During the development of cartography, technology was changing, and continues to change, in order to meet the demands of new generations of mapmakers and map users. The first maps were manually constructed with brushes and parchment and therefore varied in quality and were limited in distribution. The advent of magnetic devices, such as the compass and much later magnetic storage devices, allowed for the creation of far more accurate maps and the ability to store and manipulate them digitally.
Advances in mechanical devices such as the printing press, quadrant and vernier allowed for the mass production of maps and the ability to make accurate reproductions from more accurate data. Optical technology, such as the telescope, sextant and other devices that use telescopes, allowed for accurate surveying of land and the ability of mapmakers and navigators to find their latitude by measuring angles to the North Star at night or the sun at noon.
Advances in photochemical technology, such as the lithographic and photochemical processes, have allowed for the creation of maps that have fine details, do not distort in shape and resist moisture and wear. This also eliminated the need for engraving which further shortened the time it takes to make and reproduce maps.
In the mid to late 20th century advances in electronic technology have led to a new revolution in cartography. Specifically computer hardware devices such as computer screens, plotters, printers, scanners (remote and document) and analytic stereo plotters along with visualization, image processing, spatial analysis and database software, have democratized and greatly expanded the making of maps.

See Also

list of cartographers
Sea level

Reference

Fran Evanisko, American River College, lectures for Geography 20: "Cartographic Design for GIS", Fall 2002
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Cartography."

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Function

(From Wikipedia, the free Encyclopedia)

This article covers mathematics. Other uses of the word function include:
In sociology, social functions are the basis of functionalism.
In computer science, a function is a subprogram or subroutine, commonly one intended to directly return a value to its caller. See also functional programming.

The concept of function is fundamental in mathematics and the sciences.

Introduction
Intuitively, a function is a way to assign to each value of the argument x a unique value of the function f(x). This could be specified by a formula, a relationship, and/or a rule. This concept is deterministic, always producing the same result or output from the same input. A function may be thought of as a "machine" or "black box" converting valid input into a unique output.
The most familiar kind of function is that where the argument and the function's value are both numbers, and the functional relationship is expressed by a formula, and the value of the function is obtained from the arguments by direct substitution. Consider for example

which assigns to any number x its square.
A straightforward generalization is to allow functions depending not on a single number, but on several. For instance,

which takes two numbers x and y and assigns to them their product, xy.
In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on Earth over time: this is a function which takes location and time as arguments and gives as output the temperature at that location at that time.
We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that to every element of the domain is associated precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, functions appear in a wide variety of mathematical contexts, and several mathematical fields are based on the study of functions.
The words "function", "mapping", "map" and "transformation" are usually considered synonymous. Functions whose arguments are natural numbers are better known as sequences.

History
As a mathematical term, "'function\'" was coined by Leibniz in 1694, to describe a quantity related to a curve; such as a curve's slope or a specific point of said curve. Functions related to curves are nowaday called differentiable functions and are still the most frequently type of functions encounted by non-mathematicians. For such kind of functions, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and they are the basics of calculus.
The word function was later used by Euler during the mid-18th Century to describe an expression or formula involving various arguments; ie: y = F(x).
During the 19th Century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis).
By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as functions which are nowhere differentiable. Those functions, first thought as purely imaginary and called collectively "monsters" as late as the turn of the 20th century, were later found to be important in the modelling of physical phenomena such as Brownian motion.
Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory and they sought definitions of every mathematical object as a set. It was Dirichlet that gave the modern "formal" definition of function (see #Formal Definition below).
In Dirichlet's definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible.

Formal Definition
Formally, a function f from a set X of input values to a set Y of possibly output values (written as f: X → Y) is a relation between X and Y which satisfies:
f is functional: if x f y (x is f-related to y) and x f z, then y = z. i.e., for each input value, there should only be one possible output value.
f is total: for all x in X, there exists a y in Y such that x f y. i.e. for each input value, the formula should produce at least one output value within Y.

For each input value x in the domain, the corresponding unique output value y in the codomain is denoted by f(x).
Consider the following three examples:

This is not a "well-defined" function; because, the element 3, in X, is associated with two elements b and c in Y (Condition 1 is violated). This is a multivalued function.
This is not a "well-defined" function; because, the element 1, in X , is associated with nothing (Condition 2 is violated). This is a partial function.
This is a function, called a discrete function (or rarely piecewise function); of which the range is {a,c,d}. It can be stated explicitly as

Occasionally, all three relations above are called functions. In this case, the function satisfies Conditions (1) and (2) is said to be a "well-defined function" or "total function". In this encyclopedia, the terms "well-defined function", "total function" and "function" are synonymous.

Domains, Codomains, and Ranges
X, the set of input values, is called the domain of f and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is wrongly called the range because of a failure to distinguish between possible and actual values.
In computer science, the datatypes of the arguments and return values specify the domain and codomain (respectively) of a subprogram. So the domain and codomain are constraints imposed initially on a function; on the other hand the range has to do with how things turn out in practice.

Graph of a functions
The graph of a function f is the collection of all points(x, f(x)), for all x in set X. In the example of the discrete function, the graph of f is {(1,a),(2,d),(3,c)}. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem.
If X and Y are real lines, then this definition coincides with the familiar sense of graph. Below is the graph of a cubic function:

Note that since a relation on the two sets X and Y is usually formalized as a subset of X×Y, the formal definition of function actually identifies the function f with its graph.

Images and preimages
The image of an element x∈X under f is the output f(x).
The image of a subset A⊂X under f is the subset of Y defined by
f(A) := {f(x) : x in A}.
Notice that the range of f is the image f(X) of its domain. In our example of discrete function, the image of {2,3} under f is f({2,3})={c,d} and the range of f is {a,c,d}.
The preimage (or inverse image) of a set B ⊂ Y under f is the subset of X defined by
f⁻¹(B) := {x in X : f(x)∈B}.
In our example of discrete function, the preimage of {a,b} is f⁻¹({a,b})={1}.
Note that with this definiton, f^-1 becomes a function whose domain is the set of all subsets of Y (also known as the power set of Y) and whose codomain is the power set of X'.
Some consequences that follow immediately from these definitions are:
f(A₁ ∪ A₂) = f(A₁) ∪ f(A₂).
f(A₁ ∩ A₂) ⊆ f(A₁) ∩ f(A₂).
f⁻¹(B₁ ∪ B₂) = f⁻¹(B₁) ∪ f⁻¹(B₂).
f⁻¹(B₁ ∩ B₂) = f⁻¹(B₁) ∩ f⁻¹(B₂).
f(f⁻¹(B)) ⊆ B.
f⁻¹(f(A)) ⊇ A.

These are valid for arbitrary subsets A, A₁ and A₂ of the domain and arbitrary subsets B, B₁ and B₂ of the codomain. The results relating images and preimages to the algebra of intersection and union work for any number of sets, not just for 2.

Injective, surjective and bijective functions
Several types of functions are very useful, deserve special names:
injective (one-to-one) functions send different arguments to different values; in other words, if x and y are members of the domain of f, then f(x) = f(y) if and only if x = y. Our example is an injective function.
surjective (onto) functions have their range equal to their codomain; in other words, if y is any member of the codomain of f, then there exists at least one x such that f(x) = y.
bijective functions are both injective and surjective; they are often used to show that the sets X and Y are "the same" in some sense.

Examples of functions
(More can be found at List of functions.)

The relation wght between persons in the United States and their weights.
The relation between nations and their capitals.
The relation sqr between natural numbers n and their squares n².
The relation ln between positive real numbers x and their natural logarithms ln(x). Note that the relation between real numbers and their natural logarithms is not a function because not every real number has a natural logarithm; that is, this relation is not total and is therefore only a partial function.
The relation dist between points in the plane R² and their distances from the origin (0,0).
The relation grav between a point in the punctured plane R² \\ {(0,0)} and the vector describing the gravitational force that a certain mass at that point would experience from a certain other mass at the origin (0,0).

Most commonly used types of mathematical functions involving addition, division, exponents, logarithms, multiplication, polynomials, radicals, rationals, subtraction, and trigonometric expressions. They are sometimes collectively referred as Elementary functions -- but the meaning of this term varies among different branches of mathematics. Example of non-elementary functions are Bessel functions and gamma functions.

n-ary function: function of several variables
Functions in applications are often functions of several variables: the values they take depend on a number of different factors. From a mathematical point of view all the variables must be made explicit in order to have a functional relationship - no 'hidden' factors are allowed. Then, again from the mathematical point of view, there is no qualitative difference between functions of one and of several variables. A function of three real variables is just a function that applies to triples of real numbers. The following paragraph says this in more formal language.
If the domain of a function is a subset of the Cartesian product of n sets then the function is called an n-ary function. For example, the relation dist has the domain R × R and is therefore a binary function. In that case dist((x,y)) is simply written as dist(x,y).
Another name applied to some types of functions of several variables is operation. In abstract algebra, operators such as "*" are defined as binary functions; when we write a formula such as x*y in this context, we are implicitly invoking the function *(x,y), but writing it in a convenient infix notation.
An important theoretical paradigm, functional programming, takes the function concept as central. In that setting, the handling of functions of several variables becomes an operational matter, for which the lambda calculus provides the basic syntax. The composition of functions (see under composing functions immediately below) becomes a question of explicit forms of substitution, as used in the substitution rule of calculus. In particular, a formalism called currying can be used to reduce n-ary functions to functions of a single variable.

Composing functions
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a function g o f: X → Z defined by (g o f)(x) := g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x). Then (c o h)(t) describes the oxygen concentration around the plane at time t.
If Y⊂X then f may compose with itself; this is sometimes denoted f². (Do not confuse it with the notation commonly seen in trigonometry.) The functional powers f ofⁿ = fⁿ o f = fⁿ⁺¹ for natural n follow immediately. On their heels comes the idea of functional root; given f and n, find a g such that gⁿ=f. (Feynman illustrated practical use of functional roots in one of his anecdotal books. <which?> Tasked with building an analogue arctan computer and finding its parts overstressed, he instead designed a machine for a functional root <fifth?> of arctan and chained enough copies to make the arctan machine.)

Inverse function
If a function f:X→Y is bijective then preimages of any element y in the codomain Y is a singleton. A function taking y∈Y to its preimage f⁻¹(y) is a well-defined function called the inverse of f and is denoted by f⁻¹.
An example of an inverse function, for f(x) = x², is f(x)⁻¹ = √x. Likewise, the inverse of 2x is x/2. The inverse function is the function that "undoes" its original. See also inverse image.

Pointwise operations
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows:
(f + g)(x) := f(x) + g(x);
(f × g)(x) := f(x) × g(x);
for all x in X.
This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.
By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way.

Computable and non-computable functions
The number of computable functions from integers to integers is countable, because number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. This argument shows that there are functions from integers to integers that are not computable. For examples of noncomputable functions, see the articles on the halting problem and Rice's theorem.

References

Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Function."

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Graph theory

(From Wikipedia, the free Encyclopedia)

Graph theory is the branch of mathematics that examines the properties of graphs.

A graph with 6 vertices and 7 edges.
Informally, a graph is a set of objects called vertices or nodes connected by links called edges or arcs. Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges).
For more and formal definitions, see Glossary of graph theory and Graph (mathematics).
Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a directed graph, or digraph.
Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be formulated as questions about certain graphs. For example, the link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia, and there's a directed edge from article A to article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to handle graphs is therefore of major interest in computer science.

History
Leonhard Euler's paper on Seven Bridges of Königsberg is considered to be the first result in graph theory. It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. This illustrates the deep connection between graph theory and topology.

Graph problems

Coloring graphs: the four color theorem

Route problems:
Seven bridges of Königsberg
Minimum spanning tree
Shortest path problem
Route inspection problem (also called the "Chinese Postman Problem")
Traveling salesman problem

Flows:
Max flow min cut theorem

reconstruction conjecture

Isomorphism problems (Graph matching)
Canonical Labeling
subgraph isomorphism and monomorphisms
Maximal common subgraph

Important algorithms

Dijkstra's algorithm
Kruskal's algorithm
Nearest neighbour algorithm
Prim's algorithm

Generalizations
In a hypergraph an edge can connect more than two vertices.
An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.
Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.
In model theory, a graph is just a structure. But in that case, there is no limitations on the number of edges: it can be any cardinal number.

Related areas of mathematics

Ramsey theory
Combinatorics
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Graph theory."

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Map

(From Wikipedia, the free Encyclopedia)

zh-cn:地图
This article is about geographic maps. For maps in mathematics, see function (mathematics).
A map is (almost universally) a two-dimensional representation of a three-dimensional space.
The science of making maps is called cartography.
Early maps were vague and there was often controversy as to where to centre the map - one world map, for instance, has Jerusalem at the centre.
Many maps have a scale, determining how large objects on the map are in relation to their actual size. A larger scale shows more detail, thus requiring a larger map to show the same area. Some, though, are not drawn to scale - a famous exmmple being the London Underground map
If the map covers a large area of the surface of a globe, such as the Earth, it also has a projection, a way of translating the three-dimensional real surface of the geoid to a two-dimensional picture. The most commonly used is the Mercator Projection; other popular projections are polar and a variety of equal-area projections.
The features shown on a map vary according to its purpose. For example, a road map may or may not show railroads, and if it does, it may show them less clearly than highways.
Maps can be political or geographical. The most important purpose of the political map is to show territorial borders; the purpose of the geographical is to show features of physical geography such as mountains, soil type or land use. Geological maps show not only the physical surface, but characteristics of the underlying rock, fault lines, and subsurface structures.
Many mapping projects have been carried out by the military. An example of this the British Ordnance Survey (now a civilan government operation).
Because maps are abstract representations of the world they are not neutral documents and must be carefully interpreted.

Other uses

Road map is also used metaphorically for a plan, for example "Road map for peace".
In first person shooters and other computer games "map" refers to the current territory (including buildings, entities, spawn points, etc.) as well as the objectives that must be completed. For example, the map "de_dust" in Counter-Strike includes the brushes, textures, bomb sites, spawn points, and backgrounds.
Song Lines as maps in Australian Aborigine culture.

See also

Atlas (cartography)
Map wiki to include wiki maps in wikipedia.

References

David Buisseret, ed., Monarchs, Ministers and Maps: The Emergence of Cartography as a Tool of Government in Early Modern Europe. Chicago: University of Chicago Press, 1992, [ISBN 0226079872]
Mark Monmorier, How to Lie with Maps, [ISBN 0226534219]

External links

MapQuest: on US, Canada and Western Europe more detailed than the rest of the world
Yahoo Maps: on US, Canada, Germany, France, Spain, Italy
Yahoo Germany: on France, UK, Germany, Italy, Spain, Portugal, Austria, Switzerland, Benelux
Expedia: World atlas, partly to street level
UT scanned collection: by the University of Texas at Austin
Example of legend (Cito-Plan city maps)
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Map."

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Mapping

(From Wikipedia, the free Encyclopedia)

In mathematics and related technical fields, a mapping or map is a variation on the theme of a function.

Most commonly, the terms are simply synonyms for function. Along these lines, a partial mapping is a partial function, and a total mapping is a total function.

In many specific branches of mathematics, the term is used for a function with a standard property relevant to that branch, such as a continuous function in topology, a linear transformation in linear algebra, etc. In Wikipedia, we always include a relevant adjective like "continuous" or "smooth" to avoid confusion.

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

In computer science, a mapping is the relationship between a key and its value in an associative array.

In cognitive psychology, a mapping is the relationship between a source domain and target domain, typically reflecting a conceptual metaphor.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Mapping."

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Abbreviations & Acronyms: Map

The following table is compiled from various sources, across various languages. When English abbreviations or acronyms come from a non-English source, this is noted.
Entry	Source	Expression	Field
MAP	Dutch	Actieplan voor aspirant-leden	Military & Defense
MAP	English	Mitogen activated protein	N/A
MAP	Finnish	Valmistusautomaatioprotokolla	Post & Telecom
MAP	French	Maladie de l'Amaigrissement du Porcelet	Biology & Biotechnology, Medicine
MAP	German	Luftfahrkarten	Geography, Transportation
MAP	Italian	Carte e mappe aeronautiche	Geography, Transportation
MAP	Swedish	Autotillverkningsprotokoll	Post & Telecom
MAP 2000	English	Modernization of Administration and Personnel for 2000	N/A
MAP 2000	French	Modernisation de l'administration et du personnel pour l'an 2000	European Union
MANS	English	Map Analysis System	N/A
Source: compiled by the editor, based on several corpora (additional references).

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Synonyms: Map

Synonyms: correspondence (n), mapping (n), map out (v), represent (v). (additional references)

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Synonyms within Context: Map

Context	Synonyms within Context (source: adapted from Roget's Thesaurus).
Information	Valet de place, cicerone, pilot, guide; guidebook, handbook; vade mecum; manual; map, plan, chart, gazetteer; itinerary; (journey).
Journey	Plan, itinerary, guide; handbook, guidebook, road book; Baedeker, Bradshaw, Murray; map, road map, transportation guide, subway map.
Plan	Verb: plan,scheme, design, frame, contrive, project, forecast, sketch; devise, invent; (imagine); set one's wits to work; spring a project; fall upon, hit upon; strike out, chalk out, cut out, lay out, map out; lay down a plan; shape out a course, mark out a course; predetermine; concert, preconcert, preestablish; prepare; hatch, hatch a plot concoct; take steps, take measures.
Representation	Map, plan, chart, ground plan, projection, elevation (plan).
Situation	Topography, geography, chorography; map.
Source: adapted from Roget's Thesaurus.

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Crosswords: Map

English words defined with "map": Cabot, Cadastral map, Cadastral survey, centerfold, centrefold, characteristic, chart, Chorographer, conformal projection, conic projection, conical projection, contour, Contour line, contour map, Cotidal lines ♦ equal-area map projection, equal-area projection ♦ feature, Friezed ♦ Georama, globe ♦ Heliographic chart, Homolographic projection, homolosine projection ♦ Ichnography, isarithm, isogram, isopleth ♦ map projection, Maplike, Mapped, Mercator projection, Mercator's projection ♦ Opisometer, orthomorphic projection ♦ plane table, plot, Polyconic projection ♦ relief map, road map ♦ Sanson-Flamsteed projection, Sea chart, Sebastian Cabot, Selenographic chart, sinusoidal projection, sketch map, survey, surveying ♦ weather chart, weather map, wind rose. (references)

Specialty definitions using "map": Address List Map Review, areal map, assignment area locator map ♦ Base map, base map symbol, basic map, block locator map, block map, body of a map or chart, bookmaker, map, boundary map ♦ Census Bureau map, census map, composite map, conformal map projection, convergence map ♦ digital map, dip-corrected map, Dual Independent Map Encoding ♦ EDITOR, MAP ♦ face of a map or chart, facies map, finisher, map and chart, four colour map theorem ♦ geologic map, geomorphological map ♦ hachure map, hypsometric map ♦ image map, index map, inset map, Intensity Map, international map of the world, Inundation Map, isochore map, isochrome map, isochrone map, isogam map, isopach map, isopachous map ♦ Karnaugh map ♦ line map, linear map, locality map, locator map ♦ map feature, map grid, map inset, Map It, MAP Kinase Signaling System, map legend, map of phenomena, map point, map sheet, map spot, map spot number ♦ outcrop map ♦ paleolithologic map, panoramic-profile map, Peutingerian Map, planimetric map, procedural bump map, projected map display ♦ radiation situation map, reconnaissance map, Reference map ♦ sector map, seen area map, solid map, special job cover map, statistical map, stereometric map, stereotopographic map, structural contour map, structural map, structure map, surficial features map ♦ tectonic map, Thematic map, Topographic map ♦ vicinity map, video map, visibility area map, visibility map, visible area map ♦ water-table map, weathering map. (references)

Etymologies containing "map": Napery. (references)

Non-English Usage: "Map" is also a word in the following languages with English translations in parentheses.

Cornish (boy), Dutch (folder), Frisian (folder), Indonesian (folder), Welsh (map).

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Modern Usage: Map

Domain	Usage
Screenplays	It's kind of like when you go on vacation: you plan everything out, but then one day you make a wrong turn, or take a detour, and you end up in some crazy place you can't even find on the map, doing something you never thought you'd do. Maybe you feel a little lost while it's happening, but later you realize it was the best part of the whole trip (Threesome; writing credit: Andrew Fleming.) I could hardly find this place with a bloody map! (Sleuth; writing credit: Anthony Shaffer) I gave you the map. (The Blair Witch Project; writing credit: Daniel Myrick; Eduardo Sánchez) According to the map we've only gone 4 inches (Dumb & Dumber; writing credit: Peter Farrelly; Bennett Yellin) I have a map of every mall from here to Tijuana (Saved by the Bell; writing credit: Ana Maria Moretzsohn)
Lyrics	Every state on the map, a different somethin to eat (Po' Folks; performing artist: Nappy Roots) We'll take the trail marked on your father's map (Kiss Me; performing artist: Sixpence None The Richer)
Clever	Never trust a Private with a loaded weapon, or an Officer with a map. (references; author: unknown)
Movie/TV Titles	By Map and Compass (1972) They're Putting Us Off the Map (1968) The Impossible Map (1947) Behind the Map (1917) Off the Map (2003)
Source: compiled by the editor from various references; see credits.

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Commercial Usage: Map

Domain	Title
Books	Washington, District of Columbia: Including Alexandria Va, Annapolis MD, Arlington Va ... Rockville MD: Plus Washington DC Visitor's Map ... Toll & Fe (reference) Port Angeles/Aberdeen/ Hoquiam City Street Map (reference) Nelles Honolulu O'ahu Travel Map (reference) Reference Maps of the Islands of Hawaii: Map of O'Ahu (reference) Lonely Planet Amsterdam: City Map (City Maps Series) (reference) (more book examples)

Periodicals	911 Reference Map Set For Benton County (reference) Berkeley County Realty Map (reference) International Map Collectors Society Journal (reference) Map International Report (reference) Map News (reference) (more periodical examples)

Theater & Movies	A Map Of The World (reference) Dora the Explorer - Map Adventures (reference) Red Hot Chili Peppers - Off the Map (reference) Map of the Human Heart (reference) Map Your Way to Better Grades (reference) (more DVD examples; more video examples)

Music	Sound Map of the Hudson River (reference) A Map of the World (1999 Film) [SOUNDTRACK] (reference) New Map of World (reference) Who Put Sac on the Map (reference) (more classical music examples; more popular music examples)

High Tech	Magellan Mapsend Streets CD For Map 330 (reference) (more camera examples; more video game examples; more computer examples; more electronic examples; more software examples)
Source: compiled by the editor from various references; see credits.

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Image Slideshow: Map

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Photo Album: Map

Thumbnail	Description & Credit	Thumbnail	Description & Credit
	A dye marker on agarose gel is used to separate DNA. The smaller fragments move faster, the larger ones move slower. This separation process is used to analyyze the size of DNA fragments, to map DNA, to separate fragments of DNA to create clones. Credit: Unknown photographer/artist.		Global map showing locations of emerging infectious disease outbreaks in the 1990s.. Credit: CDC.
	Map showing AIDS Rates per 100,000 Population Reported in 1996, United States. Credit: CDC.		Mitchell Studies Map. Credit: NASA.
	Photopolarimeter-radiometer (PPR) thermal map of Ganymede's surface. Compare to the SSI image release on 08/16/96 (above) which shows nearly the same view. (The PPR map is rotated about 30 degrees to the west of the SSI image.) (Released 09/25/96). Credit: NASA.		Image, elevation map, and model Vesta's surface. Credit: NASA.
	VRML Topographic Map Generator by Dave Pape. Credit: NASA.		Map of the 39th Parallel Arc The first great geodetic arc in the western hemisphere. Credit: Coast & Geodetic Survey Historical Image Collection.
	Map showing status of triangulation as of 1937. Credit: Coast & Geodetic Survey Historical Image Collection.		Kauai map. Credit: America's Coastlines.
Source: pictures compiled by the editor from various references; see picture credits.

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Digital Photo Gallery: Map


"Map and Compass" by Luis Alves Commentary: "A map with a compass. --------------------------- Notice: You can use this image, but please send me an e-mail if you use it, I really like to know when and where it's used, thanks :-)."	"Map" by Michelle Kwajafa Commentary: "Map at the bus stop in baltimore."
Source: photographs selected by the editor, with permission from the photographers.

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Familiar Quotations: Map

Author	Quotation
Ralph Waldo Emerson	To map out a course of action and follow it to an end requires some of the same courage that a soldier needs.
William Shakespeare	Thus is his cheek the map of days outworn.
Source: compiled by the editor from various references.

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Historic Usage: Map

Author	Date	Quotation
Treaty of Versailles	1919	The boundaries described above are drawn on a German map, scale 1/100,000, attached to the present Treaty (Map No. 3). (reference)
Source: compiled by the editor from various references.

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Use in Literature: Map

Title	Author	Quote
A Grief Observed	C.S. Lewis	I thought I could describe a state; make a map of sorrow
Sylvie and Bruno Concluded	Carroll, Lewis	And there it was, a large Map of the World, spread out on the ground
Les Miserables	Hugo, Victor	On the wall was nailed, an indication sufficient to awaken the suspicion of a police officer, an old map of France under the republic
King Richard III	Shakespeare, William	I see, as in a map, the end of all.
Grapes of Wrath	Steinbeck, John	He filled the radiator, begged a map, and studied it.
Walden	Thoreau, Henry David	Or I could refer you to Ireland, which is marked as one of the white or enlightened spots on the map.
Source: compiled by the editor from various references.

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Non-Fiction Usage: Map

Subject	Topic	Quote
Health		See the map for areas where East African trypanosomiasis can be found. (references)
		As you can see by this map, rabies is most common in the eastern United States. (references)
		By 1995, scientists had produced a map of the PKD1 gene, showing all of its molecular components. (references)
Business		A wind map of Spain, using measurements recorded since the 1950s, show that there are nine regions of Spain that offer the best potential for wind energy projects. (references)
		The wind map of Argentina is outdated and incomplete, a major complaint of sector specialists, who worry about the government’s apparent lack of concern for the industry. (references)
		Much improvement has been made to the air quality at Map Tha Phut in the past few years primarily as the result of media attention and public outcry at the environmental condition of the estate. (references)
Civil Liberties	Morocco	The Government owns the official press agency, Maghreb Arab Press (MAP), and the Arabic daily newspaper, Al-Anbaa. (references)
	Togo	A religious organization must submit its statutes, a statement of doctrine, bylaws, names and addresses of executive board members, the pastor's diploma, a contract, a site map, and a description of its financial situation. (references)
	Morocco	On October 1, the Moroccan National Press Union stated that it would closely examine the case of Channel 1 TV employee Mustapha Abbasi, whom the Government suspended "just after he presented a program on all the detentions that involved activists of the Moroccan Union for Human Rights." According to MAP, on October 25 the Rabat prosecutor initiated a preliminary investigation against Ali Lemrabet, publisher of Demain magazine for publishing an article claiming that the Royal Palace in Skhirat would be sold. (references)
Economic History	Benin	A relief map of Benin shows that it has little variation in elevation (average elevation 200 meters). (references)
	India	The political map of ancient and medieval India was made up of myriad kingdoms with fluctuating boundaries. (references)
	Guyana	It will put Guyana on the map internationally and is expected to bring international media attention and business travel. (references)
Political Rights	Singapore	The electoral map was altered dramatically, with some constituencies abolished and many other constituency borders moved. (references)
Trade	India	The MAP encourages the development, maintenance, and expansion of commercial export markets for these products. (references)
	India	Market Access Program (MAP) uses funds from CCC to help U.S. producers, exporters, private companies, and other trade organizations finance both brand and generic promotional activities for U.S. agricultural, fish and forestry products. (references)
Travel	Egypt	If going to an area you do not know well, a map may help both you and the driver, who won't have one. (references)
Worker Rights	United Kingdom	Another NGO, Change, is working on a project to map out government organizations and NGO's that are combating trafficking in women globally. (references)
Source: compiled by the editor from ICON Group International, Inc.; see credits.

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Spoken Usage: Map

Speaker	Phrase(s)
Bill Maher	Well, I don't know. You know, the military did a great job overseas. That's the military. I mean, to tell your generals to point to a map of Afghanistan and say, destroy that, I think is something Al Gore could have done also.
Dennis Miller	Hey Yasser, stop buying explosives with European money and wearing a map of Israel on your headscarf.
Source: compiled by the editor from various references; see credits.

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Speeches: Map

Speaker	Term	Phrase(s)
Lyndon B. Johnson	1963-1969	Part of the American earth-not only in description on a map, but in the reality of our shores, our hills, our parks, our forests, and our mountains-has been permanently set aside for the American public and for their benefit.
Ronald Reagan	1981-1989	Had that nuclear monopoly been in the hands of the Communist world, the map of Europe--indeed, the world--would look very different today.
Source: compiled by the editor from various references.

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Usage Frequency: Map

"Map" is generally used as a noun (singular) -- approximately 92.92% of the time. "Map" is used about 3,779 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted)

Parts of Speech	Percent	Usage per 100 Million Words	Rank in English
Noun (singular)	92.92%	3,511	2,768
Lexical Verb (infinitive)	3.73%	141	26,682
Lexical Verb (base form)	2.14%	81	36,835
Noun (proper)	1.08%	41	53,521
Noun (common)	0.13%	5	157,705
Total	100.00%	3,779	N/A

Source: compiled by the editor from several corpora; see credits.

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Expression: Map

Expressions using "map": a map mounted on cloth ♦ aeronautical map ♦ aerophotogrammetric map ♦ base map ♦ base map symbol ♦ basic map ♦ Belvedere map of the Netherlands ♦ bit map ♦ body of a map ♦ body of a map or chart ♦ Cadastral map ♦ celestial map ♦ central map reference ♦ character map ♦ climatological map ♦ consult a map ♦ contig map ♦ contiguous map ♦ contour map ♦ embossment map ♦ environment map ♦ face of a map or chart ♦ four colour map theorem ♦ genetic map ♦ geomorphological map ♦ image map ♦ international map of the world ♦ isochrone map ♦ Karnaugh map ♦ key map ♦ Lambert map projection ♦ line map ♦ linear map ♦ load map ♦ locality map ♦ log map ♦ map case ♦ map collection ♦ map convergence ♦ map drawing ♦ map grid ♦ map index ♦ map into ♦ MAP Kinase Kinase Kinases ♦ MAP Kinase Signaling System ♦ Map lichen ♦ map maker ♦ map of a town ♦ map of phenomena ♦ map of the area ♦ map of the city ♦ map of the world ♦ map out ♦ map projection ♦ map reading ♦ map reference ♦ map reference code ♦ map scale ♦ map series ♦ map sheet ♦ memory map ♦ mosaic map ♦ moving map display ♦ off the map ♦ ordnance map ♦ ordnance survey map ♦ orient a map ♦ p42 MAP Kinase ♦ physical map ♦ planimetric map ♦ political map ♦ polyconic map projection ♦ printing size of a map ♦ procedural bump map ♦ projected map display ♦ put on the map ♦ questions off the map ♦ radar map ♦ radiation situation map ♦ railway map ♦ reconnaissance map ♦ relief map ♦ risk map ♦ road map ♦ rough map ♦ route map ♦ sector map ♦ seen area map ♦ short reference map ♦ situation map ♦ skeleton map ♦ sketch map ♦ spatial map ♦ special job cover map ♦ star map ♦ statistical map ♦ storage map ♦ street map ♦ structural contour map ♦ structural map ♦ structure map. Additional references.
Hyphenated Usage
	Beginning with "map": map-based, map-case, map-cases, map-compatible, map-covered, map-engraver, map-guided, map-like, map-maker, map-makers, map-making, map-ml, map-out, map-pins, map-plotting, map-pocket, map-pouches, map-printers, map-rap, map-read, map-reader, map-reading, map-reference, map-related, map-scale-dependent, map-seller, map-sheets, map-stretch, map-table, map-to-image, map-to-read, map-to-see, map-users, map-using.
	Ending with "map": F-map.
Source: compiled by the editor from various references; see credits.

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Frequency of Internet Keywords: Map

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.

Expression	Frequency per Day	Expression	Frequency per Day
map	193,391	map of ontario	3,460
yahoo map	29,545	microsoft expedia map	3,011
united state map	24,437	map of north carolina	3,004
world map	18,373	travel map	2,960
florida map	10,202	map of germany	2,958
map direction	9,359	map driving direction	2,881
road map	8,595	michigan map	2,805
map usa	8,581	city map	2,773
map blast	7,210	france map	2,771
california map	7,067	africa map	2,752
texas map	6,389	georgia map	2,429

	This is not a "well-defined" function; because, the element 3, in X, is associated with two elements b and c in Y (Condition 1 is violated). This is a multivalued function.
	This is not a "well-defined" function; because, the element 1, in X , is associated with nothing (Condition 2 is violated). This is a partial function.
	This is a function, called a discrete function (or rarely piecewise function); of which the range is {a,c,d}. It can be stated explicitly as

Map

Definition: Map

Map

Noun

Verb

Specialty Definition: Map

Specialty Definition: Cartography

Reference

Function

Examples of functions

Graph theory

Important algorithms

Generalizations

Map

See also

References

External links

Mapping

Abbreviations & Acronyms: Map

MAP

MAP

MAP

MAP

MAP

MAP

MAP

Synonyms: Map

Synonyms within Context: Map

Information

Journey

Plan

Representation

Situation

Crosswords: Map

Modern Usage: Map

Screenplays

Lyrics

Clever

Movie/TV Titles

Commercial Usage: Map

Books

Periodicals

Theater & Movies

Music

High Tech

Image Slideshow: Map

Photos:Map

Illustrations:Map

Computer Images:Map

Photo Album: Map

Digital Photo Gallery: Map

Familiar Quotations: Map

Ralph Waldo Emerson

William Shakespeare

Historic Usage: Map

Treaty of Versailles

Use in Literature: Map

A Grief Observed

C.S. Lewis

Sylvie and Bruno Concluded

Carroll, Lewis

Les Miserables

Hugo, Victor

King Richard III

Shakespeare, William

Grapes of Wrath

Steinbeck, John

Walden

Thoreau, Henry David

Non-Fiction Usage: Map

Spoken Usage: Map

Bill Maher

Dennis Miller

Speeches: Map

Lyndon B. Johnson

Ronald Reagan

Usage Frequency: Map

Expression: Map

Frequency of Internet Keywords: Map

Photos:
Map

Illustrations:
Map

Computer Images:
Map