College Algebra Exam Review 30

College Algebra Exam Review 30 - 40 1. ALGEBRAIC THEMES We...

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40 1. ALGEBRAIC THEMES We denote by Z n the set of residue classes modulo n . The set Z n has a natural algebraic structure which we now describe. Let A and B be elements of Z n , and let a 2 A and b 2 B ; we say that a is a representative of the residue class A , and b a representative of the residue class B . The class Œa C and the class ŒabŁ are independent of the choice of representatives. For if a 0 is another representative of A and b 0 another representative of B , then a ± a 0 . mod n/ and b ± b 0 . mod n/ ; therefore a C b ± a 0 C b 0 . mod n/ and ab ± a 0 b 0 . mod n/ according to Lemma 1.7.5 . Thus Œa C D Œa 0 C b 0 Ł and ŒabŁ D Œa 0 b 0 Ł . This means that it makes sense to define A C B D Œa C and AB D ŒabŁ . Another way to write these definitions is ŒaŁ C ŒbŁ D Œa C bŁ; ŒaŁŒbŁ D ŒabŁ: (1.7.1) Example 1.7.6. Let us look at another example in which we cannot define operations on classes of numbers in the same way (in order to see what the issue is in the preceding discussion). Let
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