lec0221 - CS 173: Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Siebel Center, rm 2213 Office Hours: W 9:30-11:30a
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Cs173 - Spring 2004 CS 173 Announcements Homework 6 available. Due 02/26, 8a. Midterm 1: 2/23/06, 7-9p, SC 1404. Conflict: 2/24/06, 7-9p, SC 3405 (email me!) Sections this week will be exam review. Three additional reviews: Tues, 2/21, 6-7p, SC 1129 Thur, 2/23, 9:30-10:45a, SC 1404 Thur, 2/23, 11:00a-12:15, SC 1214
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Cs173 - Spring 2004 CS 173 Mathematical Induction, an example Prove that for all n, k 2 k =1 ν = ν(ν+129(2ν+129 6
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction If P(0) and 2200 n 0 (P(0) P(1) P(n)) P(n+1) Then 2200 n 0 P(n) In our proofs, to show P(k+1), our inductive hypothesis assures that ALL of P(0), P(1), … P(k) are true, so we can use ANY of them to make the inference.
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction An example. Given n blue points and n orange points in a plane with no 3 collinear, prove there is a way to match them, blue to orange, so that none of the segments between the pairs intersect.
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction Base case (n=1): Assume any matching problem of size less than (k+1) can be solved. Show that we can match (k+1) pairs.
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction Show that we can match (k+1) pairs. Suppose there is a line partitioning the group into a smaller one of j blues and j oranges, and another smaller one of (k+1)-j blues and (k+1)-j oranges. OK!! (by IH)
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction How do we know such a line always exists? Consider the convex hull of the points: If there is an alternating pair of colors on the hull, we’re done! OK!! (by IH)
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction If there is no alternating pair, all points on hull are the same color. Notice that any sweep of the hull hits an orange point first and also last. We sweep on some slope not given by a pair of points. OK!! (by IH)
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This note was uploaded on 09/15/2008 for the course CS 173 taught by Professor Fleck@shaffer during the Spring '08 term at University of Illinois at Urbana–Champaign.

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lec0221 - CS 173: Discrete Mathematical Structures Cinda...

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