CHAPTE R o BACKGROUN D Although this course will proceed in a somewhat historical path, tracing the evolution of certain basic algebraic ideas such as number, equation, and symmetry from early Greek mathematics to the mid-19th century, we shall from the beginning freely make use of algebraic notation, which did not emerge until the time of Descartes (early 1600s), and of concepts such as set, function, equivalence relation, etc. of an even later vintage. (The termfunction was perhaps first used by Leibniz in the early 1700s.) Most, if not all, of these concepts should be familiar to you from earlier math courses-precalculus, calculus, linear algebra, and/or foundations of higher mathematics. For this reason, we provide only a very brief review of the most important notations and concepts here. If you need more examples or lengthier explanations, ask your instructor to recommend a good book to consult. We assume an intuitive notion of the concept set. Some important sets for this course are the sets Z of integers, N of natural numbers (nonnegative integers), Q of rational numbers (fractions), R of real numbers, and C of complex numbers. Definition. If D and T are two sets, then their Cartesian product is the set Dx T = {(d,t) :de D,t eT}. Perhaps the most famous and important Cartesian product is R x R = R2. If we view R = R1 geometrically as the "real number line," then R2 is the "Cartesian plane." Absolutely crucial to all that follows is a profound understanding of the concept of a function, so we shall devote some time to this now. There are many ways to think of a function. A function / may be thought of as a rule or correspondence that assigns to each object x in some specified set D a uniquely determined object f(x) in some (possibly different) set T. The set D is then called the domain of / and T is sometimes called the target set of / . The subset R = {/(*) eT :x e D) is called the range of / . Thus D could be the set of all people living in the United States on January 1,2001, T could be the set of all colors, and / could assign to each person his or her hair color on January 1, 2001, assuming this could be determined unambiguously. 7

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