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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 relation. 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 equivalence 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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 Fall '08
 EVERAGE
 Algebra, Sets

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