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EXERCISES WITHOUT SOLUTIONS Discuss the truth of the following statements. Prove the ones that are true; find a counterexample for each one of the false statements. Exercises with the symbol (*) require knowledge of calculus and/or linear algebra. General Topics 1. I. Write each of the following statements in the form "If A, then B." II. Construct the contrapositive of each statement: Every differentiable function is continuous. The sum of two consecutive numbers is always an odd number. The product of two consecutive numbers is always an even number. No integer of the form n 2 + 1 is a multiple of 7. Two parabolas having three points in common coincide. Let b and c be any two real numbers with b c, and let a be their arithmetic average, defined as a = b + c . Then b a c. p 2 ffi ffiffi 2. The number 7 is irrational. 3. The only prime of the form n 5 - 1 is 31. 4. There is a differentiable function whose graph passes through ffiffiffi the three points (-1, 0), (0, -3), and (1, 5). p 5. The reciprocal of a nonzero number of the kind z = a + b 5 , with a and b real numbers, is a number of the same kind. 6. If x is a positive real number, then x 3 x. p ffiffiffi 7. Let n be a natural number, and x a fixed positive irrational number. Then n x is always irrational. 2