These Two Weeks. Mathematical Induction Chapter 4 Lecture 10 & Lecture 13. CPRE 310 Discrete Mathematics. Mathematical Induction Problems

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1 These Two Wees CPRE 0 Discrete Mathematics Mathematical Iductio Chapter Lecture 0 & Lecture Lecture 0 Mathematical Iductio (MI) MI Problems Lecture, Review Lecture February 9, Test I A list of topics ad practice problems is posted ad also mailed to you. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Mathematical Iductio Problems Page 79-8:,,0,,,,,,, 7 Mathematical Iductio: A Way to Prove Formulas ad Other Thigs Give a iteger variable, we ca cosider a variety of properties P() that might be true or false for various values of. For istace, we could cosider P(): 7 ( ) = P(): is divisible by P(): cets ca be obtaied usig ad cois A proof by mathematical iductio shows that a give property P() is true for all itegers greater tha or equal to some iitial iteger. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Warm-up : Cosider the expressio 7 ( ) Warm-up : Cosider the property P( ): 7 ( ) =. What is 7 ( ) whe =? whe =? ( ) ( ) ( ) = ( ) ( ) = whe =? ( ) = NOTE: The umbers,, ad 7 do t appear!. What is 7 ( ) whe =? 7 ( ) = = whe =? ( ) (() ) = ( ) What is the ext-to-last term of 7 ( )?. What is P()?. What is P()?. What is P( )? = Why the colo? Why ot =? 7 ( ) = 7 (( ) ) = ( ) Or: 7 ( ) = ( ) Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 6

2 Class Exercise Mathematical Iductio: Itroductio Do the worsheet Itroductio to Mathematical Iductio. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 7 Is there some feature of the table below that esures that it ca be cotiued idefiitely? 6 7 ( ) M ( -) ( -) ( ) M ( ) If, the? Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 8 6 equal? Mathematical Iductio: Itroductio. Equality of the two expressios i ay oe row guaratees equality i the ext row. Why? I row, we have ( ) ( ). But if we assume the expressios are equal i row, the this equals ( ) = ( ). The equality is true i some row(s). Thus, the expressios will be equal i all rows. Coclude: The table ca be cotiued idefiitely. Format for Math Iductio A proof by mathematical iductio cosists of two steps: a basis step ad a iductive step. I the basis step, oe shows that the property is true for the iitial value of. I the iductive step, oe assumes that the property is true for a particular but arbitrarily chose value of the iitial value, ad oe the shows that the property is true for the ext value of. A variatio of mathematical iductio is used to prove correctess of computer algorithms. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 9 Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 0 Mathematical Iductio: Example Example: Prove that for all itegers, 7 ( ) =. Proof: Cosider the equatio 7 ( ) = the property Show that the property is true for = : Whe =, the property is the equatio =. But the left-had side (LHS) of this equatio is, ad the right-had side (RHS) is, which equals also. So the property is true for =. It s really importat to ow what the property is. Iductive Step for proof that for all itegers, 7 ( ) =. Show that itegers, if the property is true for = the it is true for = : Let be ay iteger with, ad suppose that the property is true for =. I other words, suppose that 7 ( ) =. This suppositio is called the iductive hypothesis. We must show that the property is true for =. I other words, we must show that 7 (( ) ) = ( ), or, equivaletly, we must show that 7 ( ) = ( ). Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved.

3 Iductive hypothesis: 7 ( )=. Show: 7 ( ) = ( ). Mathematical Iductio Note But the LHS of the equatio to be show is 7 ( ) = 7 ( ) ( ) by maig the ext-to-last-term explicit = ( ) by substitutio from the iductive hypothesis = ( ) by algebra, which equals the RHS of the equatio to be show. This proves that if the property is true for =, the it is true for = ad completes the proof by mathematical iductio. Whe provig a formula by mathematical iductio, it is virtually always desirable to mae the extto-last-term explicit, as was doe above. Doig so, maes it easier to see how the iductive hypothesis will apply. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Mathematical Iductio Why do the basis ad iductive steps prove that the property is true for all itegers? I other words, why do they prove that for all itegers, 7 ( ) =? Thi bac to the reasoig we used for the table. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Outlie a proof by math iductio for: L = for all itegers 0. Proof by mathematical iductio: Let the property P() be the equatio L =. Show that the property is true for = 0: We must show that 0 =. Show that for all itegers 0, if the property is true for =, the it is true for = : Let be a iteger with 0, ad suppose that [This is the L =. iductive hypothesis.] We must show that ( ) L =. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 6 Class Exercise Do the worsheet Outliig a Proof by Mathematical Iductio. Algorithmic Notatio ad Trace Tables The Euclidea Algorithm Iput: A, B [itegers with A > B 0] Algorithm Body: a := A, b := B, r := B while (b 0) r := a mod b a := b b := r ed while gcd := a Output: gcd [gcd will be the greatest commo divisor of A ad B] Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 7 Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 8

4 Algorithmic Notatio ad Trace Tables Trace the actio of the Euclidea algorithm: A 78 B 0 a b r gcd More About Summatios. = = = = ( ad are called dummy variables ). = = ( i ) = i = 0 (Chage i limits chage i terms) Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 9 Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 0 More About Summatios More About Summatios. a b = ( a b ) = = = = c a ca = = Example: a b = ( a a) ( b b) = = = ( ( = a b) a b) ( a b ) = Example: Trasform by maig the chage of variable j =. = Whe =, the j = = 0 Whe =, the j = j = = j Thus = (j ) So: = = ( j ) j = 0 Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Outlie a proof by math iductio for: is divisible by for all itegers 0. Scratch Wor for fiishig the proof that is divisible by for all itegers 0. Proof by mathematical iductio: Let the property P() be the setece is divisible by. the property Show that the property is true for = 0: We must show that 0 is divisible by. But 0 = = 0, ad 0 is divisible by because 0 = 0. Show that for all itegers 0, if the property is true for =, the it is true for = : Let be a iteger with 0, ad suppose that [the property is true for =. I other words,] suppose that] is divisible by. iductive hypothesis We must show that [the property is true for =. I other words, we must show that] is divisible by. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Iductive hypothesis: is divisible by. Wat to show: is divisible by. Worig Bacward: Wat = (some iteger) Worig Forward: Kow = r for some iteger r Experimet: = = ( ) = = ( ) Note: Each of these terms is divisible by. So: = r (where r is a iteger) = ( r) Lecture Notes Copyright 009 S.C. Kothari all rights reserved.

5 Cautio! The fact that a property is true for a large umber of cases does ot isure that it is true i all cases. Proof is eeded!! Example : Let the property P () be the setece is prime. This setece is true for itegers through 0, but ot =. Example : Let the property P () be the setece is ot a iteger. There are itegers for which this setece is false. However, the smallest positive iteger for which the setece is false is 0,69,8,,76,67,97,97,08. Tromio Problem: Itroductio Defiitio: a tromio is a group of three uit squares arraged as follows: Tromio Problem: Show that for all itegers, ay checerboard with oe square removed ca be completely covered by tromioes. Example: Lecture Notes Copyright 009 S.C. Kothari all rights reserved. Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 6 Tromio Problem: Itroductio Defiitio: a tromio is a group of three uit squares arraged as follows: Tromio Problem: Show that for all itegers, ay checerboard with oe square removed ca be completely covered by tromioes. Example: Lecture Notes Copyright 009 S.C. Kothari all rights reserved. 7