Chapter 3

Functions

Throughout this book we will want to compare sets. We do this by studying the functions that exist between them. A function from a set A into a set B is a rule that associates each element xA with exactly one element yB. If we name the rule f, then we will write

images

to denote the fact that the element x is associated with y. We will also say that y is the image of x under f. Some of you are familiar with functions from that third year of high school math. In that case a function was usually a mapping or a general rule that acted on real numbers. For example,

images

would be the function from images into images that associates each number x with its square x2. Others may have written this function as x2 = y. In either case we are using some algebra to denote the image of x.

Our functions will only occasionally operate on real numbers. The reason is that our deliberations are on general sets and not exclusively on real numbers. For example, we might define the function f that takes each person on Earth to his or her age in years, or we might define a function that associates each ...

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