Map

Resources

A map, or mapping, is a rule, often expressed as an equation, that specifies a particular element of one set for each element of another. To help understand the notion of map, it is useful to picture the two sets schematically, and map one onto the other, by drawing connecting arrows from members of the first set to the appropriate members of the second set.

For instance, let the set mapped from be well-known cities in Texas, specifically, let A = {Abilene, Amarillo, Dallas, Del Rio, El Paso, Houston, Lubbock, Pecos, San Antonio}. We will map this onto the set containing whole numbers of miles. The rule is that each city maps onto its distance from Abilene. The map can be shown as a diagram in which an arrow points from each city to the appropriate distance (Figure 1).

A relation is a set of ordered pairs for which the first and second elements of each ordered pair are associated or related. A function, in turn, is a relation for which every first element of an ordered pair is associated with one, and only one, second element. Thus, no two ordered pairs of a function have the same first element. However, there may be more than one ordered pair with

the same second element. The set, or collection, of all the first elements of the ordered pairs is called the domain of the function. The set of all second elements of the ordered pairs is called the range of the function. A function is a set, so it can be defined by writing down all the ordered pairs that it contains. This is not always easy, however, because the list may be very lengthy, even infinite (that is, it may go on forever).

When the list of ordered pairs is too long to be written down conveniently, or when the rule that associates the first and second elements of each ordered pair is so complicated that it is not easily guessed by looking at the pairs, then it is common practice to define the function by writing down the defining rule. Such a rule is called a map, or mapping, which, as the name suggests, provides directions for superimposing each member of a function’s domain onto a corresponding member of its range. In this sense, a map is a function. The words “map” and “function” are often used interchangeably. In addition, because each member of the domain is associated with one and only one member of the range, mathematicians also say that a function maps its domain onto its range, and refer to members of the range as values of the function.

The concept of map or mapping is useful in visualizing more abstract functions, and helps to remind us that a function is a set of ordered pairs for which a well defined relation exists between the first and second elements of each pair. The concept of map is also useful in defining what is meant by composition of functions. Given three sets A, B, and C, suppose that A is the domain of a function f, and that B is the range of f. Further, suppose that B is also the domain of a second function g, and that C is the range of g. Let the symbol o

KEY TERMS

Domain —The set, or collection, of all the first elements of the ordered pairs of a function is called the domain of the function.

Function —A function is a set of ordered pairs or which the first and second elements of each ordered pair are related, and for which every first element of an ordered pair is associated with one, and only one, second element.

Range —The set containing all the values of the function.

represent the operation of composition that is defined to be the process of mapping A onto B and then mapping B onto C. The result is equivalent to mapping A directly onto C by a third function, call it h. This is writte n g so f = h, and read “the composition of f and g equals h. ”

Resources

BOOKS

Christian, Robert R. Introduction to Logic and Sets. Waltham, MA: Blaisdell Publishing Co., 1965.

Gowar, Norman. An Invitation to Mathematics. New York: Oxford University Press, 1979.

Kyle, James. Mathematics Unraveled. Blue Ridge Summit, PA: Tab Book, 1976.

Peterson, Ivars. Islands of Truth, A Mathematical Mystery Cruise. New York: W. H. Freeman, 1990.

OTHER

Wolfram MathWorld. “Map” <http://mathworld.wolfram.com/Map.html> (accessed December 3, 2006).

J. R. Maddocks

Map

A map, or mapping, is a rule, often expressed as an equation, that specifies a particular element of one set for each element of another set. To help understand the notion of map, it is useful to picture the two sets schematically, and map one onto the other, by drawing connecting arrows from members of the first set to the appropriate members of the second set. For instance, let the set mapped from be well-known cities in Texas, specifically, let A = {Abilene, Amarillo, Dallas, Del Rio, El Paso, Houston, Lubbock, Pecos, San Antonio}. We will map this onto the set containing whole numbers of miles. The rule is that each city maps onto its distance from Abilene. The map can be shown as a diagram in which an arrow points from each city to the appropriate distance.

A relation is a set of ordered pairs for which the first and second elements of each ordered pair are associated or related. A function , in turn, is a relation for which every first element of an ordered pair is associated with one, and only one, second element. Thus, no two ordered pairs of a function have the same first element. However, there may be more than one ordered pair with the same second element. The set, or collection, of all the first elements of the ordered pairs is called the domain of the function. The set of all second elements of the ordered pairs is called the range of the function. A function is a set, so it can be defined by writing down all the ordered pairs that it contains. This is not always easy, however, because the list may be very lengthy, even infinite (that is, it may go on forever). When the list of ordered pairs is too long to be written down conveniently, or when the rule that associates the first and second elements of each ordered pair is so complicated that it is not easily guessed by looking at the pairs, then it is common practice to define the function by writing down the defining rule. Such a rule is called a map, or mapping, which, as the name suggests, provides directions for superimposing each member of a function's domain onto a corresponding member of its range. In this sense, a map is a function. The words map and function are often used inter-changeably. In addition, because each member of the domain is associated with one and only one member of the range, mathematicians also say that a function maps its domain onto its range, and refer to members of the range as values of the function.

The concept of map or mapping is useful in visualizing more abstract functions, and helps to remind us that a function is a set of ordered pairs for which a well defined relation exists between the first and second elements of each pair. The concept of map is also useful in defining what is meant by composition of functions. Given three sets A, B, and C, suppose that A is the domain of a function f, and that B is the range of f. Further, suppose that B is also the domain of a second function g, and that C is the range of g. Let the symbol o represent the operation of composition which is defined to be the process of mapping A onto B and then mapping B onto C. The result is equivalent to mapping A directly onto C by a third function, call it h. This is written g o f = h, and read "the composition of f and g equals h."

Resources

books

Christian, Robert R. Introduction to Logic and Sets. Waltham, MA: Blaisdell Publishing Co., 1965.

Gowar, Norman. An Invitation to Mathematics. New York: Oxford University Press, 1979.

Kyle, James. Mathematics Unraveled. Blue Ridge Summit, PA: Tab Book, 1976.

Peterson, Ivars. Islands of Truth, A Mathematical MysteryCruise. New York: W. H. Freeman, 1990.

J. R. Maddocks

KEY TERMS

Domain

—The set, or collection, of all the first elements of the ordered pairs of a function is called the domain of the function.

Function

—A function is a set of ordered pairs or which the first and second elements of each ordered pair are related, and for which every first element of an ordered pair is associated with one, and only one, second element.

Range

—The set containing all the values of the function.

map / map/ • n. 1. a diagrammatic representation of an area of land or sea showing physical features, cities, roads, etc.: a street map | fig. expansion of the service sector is reshaping the map of employment. ∎  a two-dimensional representation of the positions of stars or other astronomical objects. ∎  a diagram or collection of data showing the spatial arrangement or distribution of something over an area: an electron density map. ∎  Biol. a representation of the sequence of genes on a chromosome or of bases in a DNA or RNA molecule. ∎  Math. another term for mapping. 2. inf., dated a person's face.• v. (mapped , map·ping ) [tr.] represent (an area) on a map; make a map of: inaccessible parts will be mapped from the air. ∎  record in detail the spatial distribution of (something): the project to map the human genome. ∎  [tr.] associate (a group of elements or qualities) with an equivalent group, according to a particular formula or model: the transformational rules map deep structures into surface structures. ∎  Math. associate each element of (a set) with an element of another set. ∎  [intr.] be associated or linked to something: it is not obvious that the subprocesses of language will map onto individual brain areas.PHRASES: off the map (of a place) very distant or remote: just a hick town, right off the map.put something on the map bring something to prominence: the exhibition put Cubism on the map.wipe something off the map obliterate something totally.PHRASAL VERBS: map something out plan a route or course of action in detail: I mapped out a route over familiar country near home.DERIVATIVES: map·less adj.map·pa·ble adj.map·per n.

MAP Acronym for Manufacturing Automation Protocol. A set of protocols originally devised by a group of US manufacturers of mechanical engineering products. This original group has been expanded to include other parties, and the protocols have become ISO OSI (open systems interconnection) standards. The protocols are intended to facilitate the exchange of data relevant to mechanical-engineering design and manufacture. They cover not only the problems of process control and assembly within a single manufacturing plant, but also the exchange of design and manufacturing data between a main contractor and his subcontractors. See also TOP, STEP.

map
1. (mapping) See function.

2. See memory map.

3. See bitmap, pixmap.

4. See Karnaugh map.