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MATH 260 2001/2002 Midterm Exam 1 November 9, 2001 Dr. Bing Zhou Name Instructions. All answers should be clear and complete. Show all your works. Partial credit will be given for the part of your work that leads to a correct answer. 1. (5 points) (a) Determine whether the following statement is true or false (explain why briefly). The universe of discourse is R, the set of all real numbers. 1. ∃x∀y (y > x) . ´ ³ 2 2. ∃x∀y (x + y) = x2 + y2 . (b) Find the converse and contrapositive of the statement “x is an even number is the necessary condition for x to be a multiple of 4”. 2. (5 points) Let A be a set of eight elements and B a set of fifteen elements. How many functions from A to B are not one-to-one? 3. (5 points) Let x, y, and z be three integers such that xy = z. Prove that if x and z are odd numbers, y must be an odd number. 2 4. (5 points) Use mathematical induction to prove that (n + 1) ≤ 2n2 for all n ≥ 3. 2