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Unformatted text preview: 2.6. EQUIVALENCE RELATIONS AND SET PARTITIONS 127 Definition 2.6.2. A partition of a set X is a collection of mutually disjoint nonempty subsets whose union is X . Equivalence relations and set partitions are very common in mathe matics. We will soon see that equivalence relations and set partitions are two aspects of one phenomenon. Example 2.6.3. (a) For any set X , equality is an equivalence relation on X . Two elements x;y 2 X are related if, and only if, x D y . (b) For any set X , declare x y for all x;y 2 X . This is an equivalence relation on X . (c) Let n be a natural number. Recall the relation of congruence modulo n defined on the set of integers by a b . mod n/ if, and only if, a b is divisible by n . It was shown in Proposition 1.7.2 that congruence modulo n is an equivalence relation on the set of integers. In fact, this is a special case of the coset equivalence relation, with the group Z , and the subgroup n Z D f nd W d 2 Z g ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Sets

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