8 ABSTRACT ALGEBRA From this point of view, a function is simply a glorified two-column list, though the set D may be unlistably large. The crucial feature is that the rule that determines f(x) given x must yield the same answer no matter who is applying the rule. In your calculus courses, you have become accustomed to examples of functions with domain R or R2 in which the defining rule is given by a formula written in algebraic notation, such as x or f(x,y) = 2x+y +x2y-4. Closely related to a function / is its graph Gr(f), which is the following subset o f D x I : Gr(f) = {(x,f(x)):xeD}DxT. Clearly / and Gr(f) determine each other uniquely, and some mathematicians, who like to derive all of mathematics from an axiomatic version of set theory, will identify / and Gr(f), avoiding the undefined term rule or correspondence. These ways of thinking about functions, though logically equivalent to all others, are rather static. We shall prefer a dynamic concept of a function as a map or mapping. We shall often wish to think of / as moving the "point" P in D to the point f(P) in T. Indeed usually in this course, the sets D and T will coincide, and we shall think of / as a "deformation"of D or as a "reshuffling" or permutation of the points of D. This dynamic view is implicit in the notation / : D T. For example, we may think of the function / : R -* R given by the rule fix) = x2 as a rule for folding the real number line R = R1 onto its nonnegative half, "pinning" it together at 0 and 1, stretching out the subset (1, oo) and squeezing the points of (0, 1) closer toward 0. Likewise we may think of the function r : R2 - R2 given by the rule r(x,y) = (-y,x) as a rule for rotating the Euclidean plane R2 90° counterclockwise about the point (0, 0). With this dynamic terminology, we shall speak of a point x in the domain of the function / as a fixed point of / or as fixed by / if f(x) = x. Thus in the first example, the fixed points of / are the points 0 and 1, and in the second example, the unique fixed point of r is the point (0, 0). Exercises 0.1. Consider the function T : R2 R2 defined by the rule T(x,y) = (—x, v). Describe T geometrically as a mapping of the plane. What are the fixed points ofT? 0.2. Consider the function S : R2 - R2 defined by the rule S(x, y) = (x2, y). Describe S as a mapping of the plane and determine the fixed points of S.
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