the set of all positive rationals, and the lower oval represents the set of all negative rationals. Here’s the
mapping: any number x in the upper oval is mapped into a number y in the lower oval by taking its addi-
tive inverse (multiplying x by −1). How can we define the ordered pairs in this mapping? What is the
domain? The maximal domain? The range? The co-domain? What happens in this situation if we want to
map a negative real number to something, or if we want to map something to a positive real number?
Solution
We can define the ordered pairs (x,y) as always having the form (x,−x), where x is rational and x > 0. The
domain is the set of all positive rationals. The maximal domain is the set of all positive reals. The range is the
set of all negative rationals. The co-domain is the set of all negative reals. This mapping does not tell us how to
map a negative real number to anything. It also fails to tell us how we would map anything to a positive real.
Types of Mappings
Mathematicians have special names for different types of mappings. You should know what
these terms mean, even though they may seem strange at first! Imagine two sets of objects,
called set X and set Y. Let the variable x represent an element in set X, and let the variable y
represent an element in set Y. There are three major ways in which the elements of X can be
mapped to the elements of Y.
Injection
Figure 13-3 shows a situation in which elements of set X are mapped to elements of set Y. This
mapping has a domain that is a subset of X, and a range that is a subset of Y. Each element x
Types of Mappings 211
Domain
Range
SetX= all possible
values of variable x
SetY= all possible
values of variable y
Figure 13-3 An example of an injection. Every element
x maps into a single element y, and every
elementy maps from a single element x.
