College Algebra Exam Review 120

College Algebra Exam Review 120 - f x 00 if f.x / D f.x 00...

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130 2. BASIC THEORY OF GROUPS namely a ± H b if, and only if, aH D bH . The equivalence classes of ± H are precisely the left cosets of H in G , since a ± H b , aH D bH , a 2 bH . Example 2.6.10. (a) The equivalence classes for the equivalence relation of equality on a set X are just the singletons f x g for x 2 X . (b) The equivalence relation x ± y for all x;y 2 X has just one equivalence class, namely X . (c) The equivalence classes for the relation of congruence modulo n on Z are f Œ0Ł;Œ1Ł;:::;Œn ² g . (d) Let f W X ! Y be any map. Define x 0 ± f x 00 if, and only if, f.x 0 / D f.x 00 / . The equivalence classes for the equivalence relation ± f are the fibers of f , namely the sets f ± 1 .y/ for y in the range of f . (e) Let X be any set, and let T W X ! X be an bijective map of X . For x;y 2 X , declare x ± y if there is an integer n such that T n .x/ D y . The equivalence classes for this relation are the orbits of T , namely the sets O.x/ D f T n .x/ W n 2 Z g for x 2 X . Equivalence Relations and Surjective Maps There is a third aspect to the phenomenon of equivalence relations and partitions. We have noted that for any map f from a set X to an- other set Y , we can define an equivalence relation on X by x 0 ±
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Unformatted text preview: f x 00 if f.x / D f.x 00 / . We might as well assume that f is surjective, as we can replace Y by the range of f without changing the equivalence rela-tion. The equivalence classes of f are the bers f 1 .y/ for y 2 Y . See Exercise 2.6.1 . On the other hand, given an equivalence relation on X , dene X= to be the set of equivalence classes of and dene a surjection of X onto X= by .x/ D x . If we now build the equivalence relation associated with this surjective map, we just recover the original equiva-lence relation. In fact, for x ;x 00 2 X , we have x x 00 , x D x 00 , .x / D .x 00 / , x x 00 . We have proved the following result: Proposition 2.6.11. Let be an equivalence relation on a set X . Then there exists a set Y and a surjective map W X ! Y such that is equal to the equivalence relation ....
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