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College Algebra Exam Review 117

College Algebra Exam Review 117 - 2.6 EQUIVALENCE RELATIONS...

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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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