sols7

# sols7 - 18 7. September 20th: Relations. 7.1. Let be an...

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18 7. September 20th: Relations. 7.1 . Let ± be an order relation on a set A . Prove that if a minimum a exists for ± , then it is unique. That is, prove that if a 1 and a 2 are both minima, then necessarily a 1 = a 2 . Answer: Recall (paragraphs preceding Remark 7.7) that a is a minimum if a ± b for all b A . If a 1 and a 2 are both minima, then a 1 ± a 2 , because a 1 is a minimum; a 2 ± a 1 , because a 2 is a minimum. It follows that a 1 = a 2 , because we are assuming that ± is an order relation, and order relations are antisymmetric. ± 7.2 . Let n be a Fxed positive integer. DeFne a relation on Z by: a b ⇐⇒ a and b have the same remainder after division by n . Prove that for every given n , is an equivalence relation. Answer: We have to verify that the relation is re±exive, symmetric, transitive. Re±exive: ²or every a , a has the same remainder as itself; therefore a a . Symmetric: ²or all a, b , if a has the same remainder as b , then b has the same remainder as a . Therefore a b = b

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## This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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sols7 - 18 7. September 20th: Relations. 7.1. Let be an...

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