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Unformatted text preview: 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....
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This note was uploaded on 09/08/2011 for the course MTH 310 taught by Professor R.bell during the Spring '10 term at Michigan State University.
- Spring '10