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lec0221 - CS 173 Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren [email protected] Siebel Center, rm 2213 Office Hours: W 9:30-11:30a
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Cs173 - Spring 2004 CS 173 Announcements Homework 6 available. Due02/26, 8a. Midterm 1: 2/23/06, 7-9p, SC 1404. Conflict: 2/24/06, 7-9p, SC 3405 (email me!) Sections this week will beexam 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 + 1 29 (2 ν + 1 29 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 bluepoints and n orange points in a plane with no 3 collinear, provethereis a way to match them, blueto orange, so that none of thesegments between thepairs intersect.
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction Basecase (n=1): Assumeany 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. Supposethere is a linepartitioning the group into a smaller oneof j blues and j oranges, and another smaller oneof (k+1)-j blues and (k+1)-j oranges. OK!! (by IH) OK!! (by IH)
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction How do weknow such a linealways exists? Consider the convex hull of the points: If thereis an alternating pair of colors on the hull, we’re done! OK!! (by IH) OK!! (by IH)
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Cs173 - Spring 2004 CS 173 Strong Mathematical Induction If thereis no alternating pair, all points on hull arethesame color.
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