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Unformatted text preview: Homework 7 Solutions Kyle Chapman November 30, 2011 153.4 We seek to show that, given a partion of S into the collection S i , there is a unique equivalence relation with the property that each S i forms an equivalence class. First we will show existence. Let be defined by a b if there exists some i such that both a and b lie in S i . For reflexsivity, we know that the collection of S i cover S so for every a in S there exists some i such that a lies in S i so a a . For symmetry, it is clear that if a b then there is some i so that both a and b lie in S i so b a . Finally, for transitivity, suppose that a b and b c . Then there is some i such that a and b both lie in S i and some j so that b and c both lie in S j . Since b lies in both S i and S j , they must be equal, so a and c both lie in S i . Thus, a c . We now know that is an equivalence relation. Next we show that its equivalence classes are the collection of S i . We know that the equivalence classes form a partition, so it suffices to show that if an....
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