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Exercises without Solutions > General Topics - Pg. 192

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