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Unformatted text preview: 11COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 4.1, pp. 127142Intro to InductionVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen214.1 Mathematical Induction•Smallest Counterexamples•The Principle of Mathematical Induction•Strong Induction•Induction in General31•We start by reviewingproof by smallest counterexampleto try and understand what it is really doing32•We start by reviewingproof by smallest counterexampleto try and understand what it is really doing•This will lead us to transform theindirect prooftechniqueofproof by counterexampletodirect prooftechnique.This direct proof technique will beinduction33•We start by reviewingproof by smallest counterexampleto try and understand what it is really doing•This will lead us to transform theindirect prooftechniqueofproof by counterexampletodirect prooftechnique.This direct proof technique will beinduction•We conclude by distinguishing between theweak principle of mathematical inductionand thestrong principle of mathematical induction34•We start by reviewingproof by smallest counterexampleto try and understand what it is really doing•This will lead us to transform theindirect prooftechniqueofproof by counterexampletodirect prooftechnique.This direct proof technique will beinduction•We conclude by distinguishing between theweak principle of mathematical inductionand thestrong principle of mathematical induction•Note that thestrong principlecan actually be derived from theweak principle. The difference between them has less to do with the power of thetechniques, than with proof format41Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by42Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a nonzero counterexample exists, i.e.,There is somen >for whichP(n)is not true2345143Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a nonzero counterexample exists, i.e.,There is somen >for whichP(n)is not true(ii) Lettingm >besmallestvalue for whichP(m)is not true2345m11mP(m)true;≤m< mP(m)not true44Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a nonzero counterexample exists, i.e.,There is somen >for whichP(n)is not true(ii) Lettingm >besmallestvalue for whichP(m)is not true(iii) Then use fact thatP(m)is true for all≤m< mto show thatP(m)is true,contradictingoriginal choice ofm.⇒P(n)true foralln= 0,1,2,...2345m11mP(m)true;≤m< mP(m)not trueP(m)true45Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a nonzero counterexample exists, i.e.,There is somen >for whichP(n)is not true(ii) Lettingm >besmallestvalue for whichP(m)is not true(iii) Then use fact thatP(m)is true for all≤m< mto show thatP(m)is true,contradicting...
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 Spring '10
 M.J.Golin
 Computer Science, Mathematical Induction, Natural number, smallest counterexample

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