# hw7-sol - CS 173 Discrete Structures Spring 2010 Homework 7...

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CS 173: Discrete Structures, Spring 2010 Homework 7 Solutions This homework contains 4 problems worth a total of 40 regular points. It is due on Friday, March 19th at 4pm. Put your homework in the appropriate dropbox in the Siebel basement. 1. Big-O proofs [10 points] For both parts of this problem, make sure your proof is presented clearly, in forward order, and spells out the (maybe 4-5 steps of) algebra clearly. (a) Prove that 3 x is O ( x 2 x +2 ). Solution: We need to prove that for some c and k that | 3 x | < c | x 2 x +2 | for all x > k . Let k = 9 and c = 4 . | 3 x | < | 3 x + ( x - 8) | = 4 | x - 2 | = 4 | x - 2 x +4 x +2 | < 4 | x - 2 x x +2 | = 4 | x 2 x +2 | This inequality proves that | 3 x | < 4 | x 2 x +2 | for all x > 9 . Alternate Solution: Without loss of generality we may assume k > 0. Since we always take x > k this allows us to ignore absolute values and avoid multiplication and division by negative numbers and zero. Observe that for all x k the following inequalities are equivalent (but we are NOT assuming them to be true yet): 3 x c · x 2 x + 2 3 x ( x + 2) c · x 2 3( x + 2) c · x 3 x + 6 c · x So to prove the ﬁrst inequality holds for all x k , it suﬃces to choose c and k so that the last inequality is satisﬁed for all x k . Choosing c = 6 and k = 2 obviously works. (b) Use a proof by contradiction to show that

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## This note was uploaded on 09/21/2011 for the course CS 173 taught by Professor Erickson during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw7-sol - CS 173 Discrete Structures Spring 2010 Homework 7...

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