412notes2 - Chapter 2, Congruence in Z and modular...

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Unformatted text preview: Chapter 2, Congruence in Z and modular arithmetic. This leads us to an understanding of the kernels and images of functions between rings (ideals, quotient rings, ring homomorphisms). It will also give us more examples of rings to think about. Definition. An equivalence relation is a binary relation which is reflexive, symmetric and transitive. Note that an equivalence relation on a set S partitions the set into subsets; these are called the equivalence classes . The application of this to congruence of integers is Theorem 2.3. (See Appendix D.) Examples: =, congruence and similarity of triangles, congruence and similarity of ma- trices and our real interest: Definition, p. 24. Let a, b, n Z with n > 0. We say a is congruent to b modulo n (written a b (mod n )) if n divides b- a . Theorem 2.1. Congruence of integers is an equivalence relation. Definition, p. 26. The equivalence class of an integer a under the relation of congruence modulo n is called the congruence class of a modulo n and denoted by [ a ]. Example. [ a ] = { b Z | b a (mod n ) } = { a + kn | k Z } Modulo 2, there are two classes: [0], the set of even numbers and [1], the set of odd numbers. Corollary 2.5. Fix n > 1 . (1) If r is the remainder when a is divided by n , then [ a ] = [ r ] (2) There are n distinct congruence classes, [0] , [1] , . . . , [ n- 1] . Proof. (1) If a = qn + r , then n divides a- r , so a r (mod n )....
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412notes2 - Chapter 2, Congruence in Z and modular...

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