Mapping, Mathematical

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Mapping, Mathematical

A mapping is a function that is represented by two sets of objects with arrows drawn between them to show the relationships between the objects.

In all mappings, the oval on the left holds values for the domain , and the oval on the right holds values for the codomain .

The figure shows a mapping of people and the sports they play. The mapping in (a) shows that Albert and Bill play basketball, Carla plays golf, and nobody plays football or hockey. The domain is {Albert, Bill, Carla}, and the codomain is {Basketball, Football, Golf, Hockey}. The sports that they do play {Basketball, Golf} comprise the range. The domain, codomain, and range are always listed using set notation .

In (b) of the figure, the domain is {Donna, Elisha, Fernando}, the codomain is {Basketball, Golf, Hockey}, and the range is {Basketball, Golf, Hockey}. In this example, the codomain and the range are the same.

Formally defined, the domain is the set of values that can be assumed by the independent variable, and the range is the set of values that can be assumed by the dependent variable. In (a) and (b), the sport played depends on the person, so sports are the dependent variable and people are the independent variable.

One-to-One and Onto Functions

Some special types of mappings are the mappings of one-to-one and onto functions. A one-to-one correspondence is a type of function in which every object in the range is paired with, at most, one object from the domain. In other words, an object in the codomain can have no more than one arrow pointing to it. Example (b) is one-to-one, but example (a) is not one-to-one because both Albert and Bill play basketball; that is, basketball has two arrows pointing to it. Example (c) is the same as (b) except it also has football in the codomain; hence, it is a one-to-one correspondence because no element of the range has more than one arrow pointing to it.

An onto function has a relationship in which every object in the codomain is paired with at least one object in the domain. This means that one or more arrows must be pointing to every value in the codomain. In other words, the codomain and the range contain the same set of objects. Example (d) is an onto function. Example (b) is also an onto function: it is a special case because it is both one-to-one and onto. The diagram in (e) summarizes how examples (a) through (d) can be classified.

Mappings for Ordered Pairs and Beyond

Mappings often are used to describe sets of ordered pairs in which the domain is the set of all x -coordinates and the range is the set of all y -coordinates. A common term used to describe mappings is image. An image of an object in the domain is the y -value that is paired with it. In (f) of the figure, the image of −3 is 1, the image of 2 is −4, and the image of 5 is 1.

Note that (f) is onto because every element of the range has arrows pointing to it, but it is not one-to-one because one element of the range has two arrows pointing to it.

Mappings for equations often represent infinite sets of ordered pairs, and therefore must be drawn a little differently. The domain represents the set of input values (x -coordinates), and the range represents the set of output values (y -coordinates). The codomain may or may not be a larger set of numbers than the range.

Example (g) of the figure illustrates how the function y = x ² is used to map from the domain, R to the codomain, R. Note that R is a symbol used to represent the set of all real numbers. Any real number can be input into the equation y = x ²; however, the output is always positive. So although the codomain is the set of all real numbers, the range is the set of positive real numbers.

Mappings can also be used to represent multidimensional problems such as mapping the ordered pairs of the vertices of a polygon to the corresponding ordered pairs for its reflection. They can also represent assigning a unique number to the coordinates of a three-dimensional object. Finally, mappings can represent n -dimensional spaces.