MATH135_F11_Assignment_6 _Solutions

MATH135_F11_Assignment_6 _Solutions - 1 MATH 135 F 2011...

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1 MATH 135 F 2011: Assignment 6 Solutions Due: 8:30 AM, Wednesday, 2011 Nov. 9 in the dropboxes outside MC 4066 Write your answers in the space provided. If you wish to typeset your solutions, use the solution template posted on the course web site. Typesetting is done in L A T E X. Cite any proposition or deﬁnition you use. Family Name: First Name: I.D. Number: Section: Mark: (For the marker only.) If you used any references beyond the course text and lectures (such as other texts, discussions with colleagues or online resources), indicate this information in the space below. If you did not use any aids, state this in the space provided. 1. A student is asked the following question: Prove that for all n P ,n 2 n - 1 . (a) A Proof by Induction will require you to show that P (1) is true. State the desired P (1). Solution: 1 2 0 . (b) Show that P (1) is true. Solution: Since 1 = 2 0 , it is also true that 1 2 0 . (c) A Proof by Induction will require you to assume that P ( k ) is true. State P ( k ). Solution: k 2 k - 1 . (d) A Proof by Induction will require you to prove that P ( k +1) is true. State the desired P ( k +1). Solution: k + 1 2 k . (e) Rewrite this incorrect “proof” to correctly show that P ( k ) P ( k + 1). Student Proof k = 2 k (By assumption) k + 1 = 2 k + 1 (Add one to both sides) = 2 k + 2 k (When k 1 , 1 2 k ) = 2 k +1 Thus k = 2 k +1 .

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2 Solution: Correct Proof k 2 k - 1 (By assumption) k + 1 2 k - 1 + 1 (Add one to both sides) 2 k - 1 + 2 k - 1 (When k
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MATH135_F11_Assignment_6 _Solutions - 1 MATH 135 F 2011...

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