Relations, Part II:
Functions and
Mappings
CHAPTER 8
At this point you may have some preconceived ideas about functions, re-
lating especially to the mechanics of working with functions and to vari-
ous purposes for which functions are used. Precalculus and calculus-level
treatment of functions, however, provide little clue to some of their uses
in higher-level mathematics (e.g., cardinality of sets, discussed in Article
8.3). Furthermore, introductory-level coverage is often so imprecise that
many students do not have a clear idea of what a function is, even though
they may know a lot about functions and, in particular, have the ability
to "know one when they see one." In this chapter we attempt to fill some
gaps in this area, and lay the groundwork for important areas of advanced
mathematics. The material covered here is fundamental to courses such as
abstract algebra, advanced calculus, and elementary topology.
We deal first with basic issues, including a precise definition of function.
/ Next, we take a second look at familiar ideas such as one-to-one function,
/ composition of functions, and inverse function. Then. we launch into new
material, including the concepts of onto mapping and one-to-one correspon-
dence between sets, and the ideas of image and inverse image of a set under
a mapping. We conclude the chapter with an introduction to cardinality
of sets and a brief consideration of arbitrary collections of sets.
8.1 Functions and Mappings
A function can be defined as a certain kind of relation. Thus a function is,
first of all, a set. More specifically, it is a set consisting of ordered pairs of
objects. The additional property that distinguishes functions, among all
relations, is specified in the following definition.