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THE FIRST MIDTERM EXAMINATION WILL BE IN CLASS ON THURSDAY, AUGUST 14 . It will cover the material through Homework IV. Mathematics 109 August 11, 2008 SAMPLE FIRST MIDTERM EXAMINATION Use no notes or books. Use no calculators or other electronic devices. Consult no one other than the Teaching Assistant or the Instructor. Write clear, careful solutions. Do all 5 problems. If a question is unclear or incorrect, please ask about it. 1. The following statements have the set of positive real numbers as their set of meanings. In each case, first write the statement without logic symbols and second either prove the statement or give a counterexample. (a) x yxy 1 (b) x yxy 1 2. This problem builds up to showing that there is no rational number whose cube is 2. (a) Show that if 2 divides n , then 8 divides n 3 . (b) Show that if 2 divides n 3 , then 2 divides n . (c) Show that there is no rational number whose cube is 2 . 3. For each part of the previous problem, indicate the method of proof used. 4. Consider sets A and B . Prove that A B A if and only if B A . 5. Prove that if each of x , y and z is a real number, then x 2 y 2 z2 xy xz yz .