Unformatted text preview: X . The equivalence classes of ± are nonempty and have union equal to X , since for each x 2 X , x 2 ŒxŁ . Furthermore, the equivalence classes are mutually disjoint ; this means that any two distinct equivalence classes have empty intersection. So the collection of equivalence classes is a partition of the set X . Any equivalence relation on a set X gives rise to a partition of X by equivalence classes. On the other hand, given a partition P of X , we can deﬁne an relation on X by x ± P y if, and only if, x and y are in the same subset of the partition. Let’s check that this is an equivalence relation. Write the partition P as f X i W i 2 I g . We have X i ¤ ; for all i 2 I , X i \ X j D ; if i ¤ j , and [ i 2 I X i D X . Our deﬁnition of the relation is x ± P y if, and only if, there exists i 2 I such that both x and y are elements of X i ....
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 Fall '08
 EVERAGE
 Algebra, Sets, Empty set, Equivalence relation

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