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We say simply that the three sets are isomorphic. 1. 2, m þ ðn þ 1Þ ¼ m þ nà ¼ ðm þ nÞà ¼ ðm þ nÞ þ 1 and P(1) is true. Next, suppose that for some k 2 N, PðkÞ : m þ ðn þ kÞ ¼ ðm þ nÞ þ k is true. We need to show that this ensures PðkÃ Þ : m þ ðn þ kÃ Þ ¼ ðm þ nÞ þ kà is true. By (ii), m þ ðn þ kÃ Þ ¼ m þ ðn þ kÞà ¼ ½m þ ðn þ kފà ðm þ nÞ þ kà ¼ ½ðm þ nÞ þ kŠÃ and Then, whenever P(k) is true, m þ ðn þ kÞà ¼ ½m þ ðn þ kފà ¼ ½ðm þ nÞ þ kŠÃ ¼ ðm þ nÞ þ kà and PðkÃ Þ is true. Thus P(p) is true for all p 2 N and, since m and n were any natural numbers, A3 follows.

Let R be an equivalence relation on S and define for each p 2 S, Tp ¼ [p] ¼ {x : x 2 S, xRp}. Since p 2 [p], it is clear that S is the union of all the distinct subsets Ta , Tb , Tc , induced by R. Now for any pair of these subsets, as Tb, and Tc, we have Tb \ Tc ¼ ; since, otherwise, Tb ¼ Tc by Problem 5. Thus, fTa , Tb , Tc , . g is the partition of S effected by R. Conversely, let fTa , Tb , Tc , . g be any partition of S. On S define the binary relation R by p R q if and only if there is a Ti in the partition such that p, q 2 Ti.

B) whenever more convenient, we may replace one system by any other isomorphic with it. Examples of this will be met with in Chapters 4 and 6. 10 PERMUTATIONS Let S ¼ f1, 2, 3, . . , ng and consider the set Sn of the n! permutations of these n symbols. A permutation of a set S is a one-to-one function from S onto S. ) The definition of the product of mappings in Chapter 1 leads naturally to the definition of a ‘‘permutation operation’’  on the elements of Sn. First, however, we shall introduce more useful notations for permutations.

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