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Unformatted text preview: 2.6. EQUIVALENCE RELATIONS AND SET PARTITIONS 129 Now for all x 2 X , x P x because there is some i 2 I such that x 2 X i . The definition of x P y is clearly symmetric in x and y . Finally, if x P y and y P z , then there exist i 2 I such that both x and y are elements of X i , and furthermore there exists j 2 I such that both y and z are elements of X j . Now i is necessarily equal to j because y is an element of both X i and of X j and X i \ X j D ; if i j . But then both x and z are elements of X i , which gives x P z . Every partition of a set X gives rise to an equivalence relation on X . Suppose that we start with an equivalence relation on a set X , form the partition of X into equivalence classes, and then build the equivalence relation related to this partition. Then we just get back the equivalence relation we started with! In fact, let be an equivalence relation on X , let P D f OEx W x 2 X g be the corresponding partition of X into equivalence classes, and let P denote the equivalence relation derived from...
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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