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 *x* ∈ *A* with exactly one element *y* ∈ *B*. If we name the rule *f*, then we will write

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,

would be the function from into that associates each number *x* with its square *x ^{2}*. Others may have written this function as

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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