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Winter 2006 - FitzGerald's Class - Exam 1

# Winter 2006 - FitzGerald's Class - Exam 1 - MATHEMATICS 109...

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MATHEMATICS 109 NAME: _______________________ February 1, 2006 Section: _______________________ FIRST MIDTERM EXAMINATION There should be no books and no notes. There should be no calculators and no cell phones. Give complete proofs written in complete English sentences, as far as practical. If a question is unclear or is incorrect, please ask the TA or Instructor about it. Do all four problems. 1. (30%) For this problem, the universe of discussion is the set of real numbers. Consider the following statement: ( 29 ( 29 ( 29 xy y x y x 2 2 2 + 2200 5 . (a) Using logical symbols write the denial of the statement a few times, each time moving the negation sign farther to the right until the final statement has no negation in it. ( 29 ( 29 ( 29 [ ] xy y x y x 2 2 2 + 2200 5 ¬ ( 29 ( 29 ( 29 [ ] xy y x y x 2 2 2 + 2200 2200 ¬ ( 29 ( 29 ( 29 [ ] xy y x y x 2 2 2 + 5 2200 ¬ ( 29 ( 29 ( 29 xy y x y x 2 2 2 + 5 2200 (b) Write the original statement and the last negation in English. Original: There exists a real number x such that for every real number y , x 2 + y 2 > 2 xy .

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Winter 2006 - FitzGerald's Class - Exam 1 - MATHEMATICS 109...

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