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 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 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."
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