Chapter VII Functions
Chapter VII Functions Greatness is not a function of circumstance. Greatness, it turns out, is largely a matter of conscious choice, and discipline. James C. Collins Functions play an important role in mathematical applications to physics, engi- neering, biology, economics, etc. If you have taken algebra and calculus, then you evaluated functions, graphed functions, and determined derivatives and integrals of functions. However, the precise definition of what a function is may have been some- what vague. In this chapter we will make precise exactly what modern mathematicians mean by a function. The definition will allow us to study functions on sets much more general than real numbers. We will see examples of functions defined on the real numbers, the integers, the natural numbers, congruence classes, power sets, and other sets. We will also discuss various properties of functions. The properties that we study have been chosen for their usefulness; they show up throughout higher mathematics. As you progress in mathematics, you will find that an important tool in studying any class of mathematical objects is the study of functions between the objects. The definitions and techniques that we learn in this chapter will prepare you for this study. 175 176 CHAPTER VII. FUNCTIONS 24 Defining functions The main purpose of this section is to formally define what we mean by a function. We will also give examples. 24.A What is a function? Recall that we have defined a relation from a set A to a set B to be a subset of A B. We now define a function from A to B as a relation that satisfies certain conditions.× Definition 24.1.
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