A the set of all subsets of the natural numbers is

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(a) The set of all subsets of the natural numbers is countable. False. (b) The number of one-to-one functions from a set with 8 elements to a set with 15 elements is P(15,8). True. (c) The coeffient of x 3 y 5 in the expansion of (x + y) 8 is the number of 3 element subsets of an 8 element set. True. (d) There is a one-to-one correspondence between the natural numbers and the real numbers in the interval (0,1). False. (e) The total number of ways to assign truth values to five true-false problems by using the letters T and F is given by the sum below. True. 5 C(5,k) k=0 6. (10 pts.) What is the minimum number of students required in your Discrete Mathematics class to be sure that at least 4 have birthdays occurring on the same day of the week this year? Explain. From the generalized pigeonhole principle, we need to find the smallest positive integer N so that N/7 = 4. Thus, N = (3*7)+1=2 2 will do. 7. (10 pts.) (10 pts.) The following proposition represents an invalid argument form: [¬p (p q)] ¬q. (a) What is the name of the popular fallacy given by the proposition above? This is the infamous fallacy of denying the hypothesis. [Some logic texts call this the fallacy of denying the antecedent.] (b) Why is reasoning using this argument form deemed invalid?
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