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Lecture 7 Questions 1. Consider the following ‘sentences’: n = 2q and m = 2r. n + m = 2q + 2r = 2(q + r). Thus n + m is even. What is the author trying to say? How should it be written? 2. For each odd number n, compute n2 − 1. Obviously, do this for only the first few odd numbers. What pattern do you see? Confirm your pattern with a classmate. Then together, try to see if you can prove that your pattern is correct. Which of the following should be regarded as defining functions? (a) f (x) = x2 − 2. √ (b) f (x) = ± x. 1 if x is rational (c) f (x) = 0 if x is irrational (d) f (~u, ~v ) = ~u × ~v . (e) f (x) = max x where x is a finite list of real numbers. R1 (f) I(f ) = 0 f (x)dx, where f (x) is continuous on [0, 1]. 3. Describe the elements of the following sets using either the roster method, or set builder notation: (a) The set consisting of the Simpsons’ family. (b) The set of all even integers. (c) The set of all integers which are the sum of two primes. (d) The days of the week. 4. Which of the following sets are subsets of other sets? (a) The set of all even numbers. (b) The set of al odd numbers. (c) The set of all multiples of three. (d) The set of all multiples of six. (e) The set of all prime numbers. 1