hw7-sol - CS 173: Discrete Structures, Spring 2010 Homework...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

hw7-sol - CS 173: Discrete Structures, Spring 2010 Homework...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online