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College Algebra Exam Review 120

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

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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 OE0Ł; OE1Ł; : : : ; OEn 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
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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 ﬁbers f ± 1 .y/ for y 2 Y . See Exercise 2.6.1 . On the other hand, given an equivalence relation ± on X , deﬁne X= ± to be the set of equivalence classes of ± and deﬁne a surjection ± of X onto X= ± by ±.x/ D ŒxŁ . 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 , Œx Ł D Œx 00 Ł , ±.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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