L10_IntroInduction

L10_IntroInduction - 1-1COMP170Discrete Mathematical...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 4.1, pp. 127-142Intro to InductionVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-14.1 Mathematical Induction•Smallest Counterexamples•The Principle of Mathematical Induction•Strong Induction•Induction in General3-1•We start by reviewingproof by smallest counterexampleto try and understand what it is really doing3-2•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 beinduction3-3•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 induction3-4•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 de-rived from theweak principle. The difference be-tween them has less to do with the power of thetechniques, than with proof format4-1Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by4-2Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a non-zero counterexample exists, i.e.,There is somen >for whichP(n)is not true234514-3Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a non-zero counterexample exists, i.e.,There is somen >for whichP(n)is not true(ii) Lettingm >besmallestvalue for whichP(m)is not true2345m-11mP(m)true;≤m< mP(m)not true4-4Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a non-zero 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,...2345m-11mP(m)true;≤m< mP(m)not trueP(m)true4-5Proof by smallest counterexamplethatstatementP(n)is true for alln= 0,1,2...works by(i) Assuming that a non-zero 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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L10_IntroInduction - 1-1COMP170Discrete Mathematical...

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