2.6. EQUIVALENCE RELATIONS AND SET PARTITIONS127Definition 2.6.2.Apartitionof a setXis a collection of mutually disjointnonempty subsets whose union isX.Equivalence relations and set partitions are very common in mathe-matics. We will soon see that equivalence relations and set partitions aretwo aspects of one phenomenon.Example 2.6.3.(a)For any setX, equality is an equivalence relation onX. Twoelementsx; y2Xare related if, and only if,xDy.(b)For any setX, declarexyfor allx; y2X.This is anequivalence relation onX.(c)Letnbe a natural number.Recall the relation ofcongruencemodulondefined on the set of integers byab .modn/if, andonly if,abis divisible byn. It was shown in Proposition1.7.2that congruence modulonis an equivalence relation on the setof integers. In fact, this is a special case of the coset equivalencerelation, with the groupZ, and the subgroupnZD fndWd2Zg.
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