# B give a recursive definition of the sequence a n

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(b) Give a recursive definition of the sequence {a n }, where n = 1,2,3,.. if a n = 2n + 1. [Be very careful of the quantification used in the induction step.] Basis Step: Induction Step: (c) Give a recursive definition of the set S of positive integers that are multiples of 3. Basis Step: Induction Step: 4. (10 pts.) Let {a n } be defined by the formula a n = 5n + 2 for n = 1,2,3, .... Define the sequence {b n } recursively by b 1 = 7 and b n+1 = b n + 5 for n = 1,2,3, .... Give a proof by induction that a n = b n for n = 1,2,3, ....

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TEST-2A/MAD2104 Page 3 of 4 5. (15 pts.) Label each of the following assertions with "true" or "false". Be sure to write out the entire word. (a) The set of all subsets of the natural numbers is countable. (b) The number of one-to-one functions from a set with 8 elements to a set with 15 elements is P(15,8). (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. (d) There is a one-to-one correspondence between the natural numbers and the real numbers in the interval (0,1). (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. 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.
TEST-2A/MAD2104 Page 4 of 4 7. (10 pts.) (10 pts.) The following proposition represents an invalid argument form: [¬p (p q)] ¬q.
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