s6 - Selected solutions to Homework 6 1#2 in Appendix D...

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Selected solutions to Homework # 6 1. #2 in Appendix D. Define a relation on Q via r s if and only if r - s Z . Prove that is an equivalence relation. Proof: We need to show three properties hold: 1. Reflexive: show that r r for every r Q . This is true since r - r = 0 Z . 2. Symmetric: show that if r s , then s r . Suppose that r s . Thus r - s Z . If a Z , then - a Z . So, - ( r - s ) = s - r Z . Therefore, s r . 3. Transitive: show that if r s and s t , then r t . Suppose that r s and s t . Then ( r - s ) and ( s - t ) are integers. This implies that ( r - s ) + ( s - t ) is an integer. Therefore r - t Z and so r t . Having checked the defining properties, we have shown that is an equivalence relation. 2. #20 in Section 1.3 We discussed this problem in class on 01/28/11. 3. #2 in Section 2.1 (a) We discussed this problem in class on 01/26/11. (b) If r 3(mod 10) and s ≡ - 7(mod 10), then what is 2 r + 3 s congruent to modulo 10?
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