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**Extra info for Schaum's Outline of Abstract Algebra (2nd Edition) (Schaum's Outlines Series)**

**Example text**

We say simply that the three sets are isomorphic. 1. 2, m þ ðn þ 1Þ ¼ m þ nÃ ¼ ðm þ nÞÃ 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 kÃ is true. By (ii), m þ ðn þ kÃ Þ ¼ m þ ðn þ kÞÃ ¼ ½m þ ðn þ kÞÃ ðm þ nÞ þ kÃ ¼ ½ðm þ nÞ þ kÃ and Then, whenever P(k) is true, m þ ðn þ kÞÃ ¼ ½m þ ðn þ kÞÃ ¼ ½ðm þ nÞ þ kÃ ¼ ðm þ nÞ þ kÃ 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.