lec0216 - CS 173: Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 9:30-11:30a
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Cs173 - Spring 2004 CS 173 Announcements Homework 5 available.  Due 02/19, 8a. Midterm 1:  2/23/06, 7-9p, SC 1404. Conflict: 2/24/06, 7-9p, SC 3405 (email me!) Sections next 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 CS173 Mathematical Induction One rule: Due to peer pressure, if the person “before” you likes  Yucky Charms, then you like Yucky Charms. Person 1 likes Yucky Charms. What can we conclude? Everyone likes Yucky Charms! Spose we want to prove everyone likes Fruit Loops Need to show two things: Suggests a proof technique Person 1 likes Fruit Loops (FL(1)) If person k likes Fruit Loops, then person k+1 does  too. (FL(k)   FL(k+1)) 2200 n FL(n)
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Cs173 - Spring 2004 CS173 Mathematical Induction Spose we want to prove everyone likes Fruit Loops Need to show two things: Person 1 likes Fruit Loops (FL(1)) If person k likes Fruit Loops, then person k+1 does  too. (FL(k)   FL(k+1)) 2200 n FL(n) First part is a simple proposition we call the base case. Second part is a conditional. Start by assuming FL(k), and show  that FL(k+1) follows. True by “peer  pressure”
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Cs173 - Spring 2004 CS173 Mathematical Induction Use induction to prove that the sum of the first n odd integers is  n 2 . Prove a base case (n=1) Base case (n=1): the sum of the first 1 odd integer is 1 2 .  Yup, 1 =  1 2 . Prove P(k) P(k+1) Assume P(k): the sum of the first k odd ints is k 2 . 1 + 3 + … + (2k -  1) = k 2 Prove that 1 + 3 + … + (2k - 1) + (2k + 1) = (k+1) 2 Inductive  hypothesis 1 + 3 + … + (2k-1) + (2k+1) = k 2  + (2k + 1) By inductive  hypothesis = (k+1) 2 By arithmetic
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Cs173 - Spring 2004 CS173 Mathematical Induction
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lec0216 - CS 173: Discrete Mathematical Structures Cinda...

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