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COMP 170 L2 Page 1

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COMP 170 L2 Page 2

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COMP 170 L2 L10: Intro to Induction l Objective n Introduce induction from proof-by-smallest-counter-example Making use of small-problem/big-problem relationships in proofs Page 3

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COMP 170 L2 Recap: 2010-03-30 l Given: (a) p(0) is True; (b) If n>0, then p(n-1) => p(n) l Conclusion: p(n) is True for all l Proof by Smallest Counter Example Page 4

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COMP 170 L2 Outline l Weak induction l Strong induction Page 5

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COMP 170 L2 Weak Principle of Math Induction l We have actually proved: Page 6

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COMP 170 L2 Weak Principle of Math Induction l Suppose b=0 l Intuitively n p(0) n p(0) => p(1) n p(1) => p(2) n p(2) => p(3), n …. n So, p(n) is True for all n>=0 Page 7

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COMP 170 L2 Inductive Proof l Base Case (n=b): Show that p(b) is True l Induction (n>b): Show p(n-1) => p(n) n Induction Hypothesis p(n-1) is True n Inductive Step: p(n) is True l Inductive conclusion: n p(n) is True for all n>=b Page 8

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COMP 170 L2 Page 9

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COMP 170 L2 Page 10

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COMP 170 L2

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Outline l Weak induction l Strong induction Page 12

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COMP 170 L2 Review: Weak Induction l Suppose b=0 l Intuitively n p(0) n p(0) => p(1) n p(1) => p(2) n p(2) => p(3), n …. n So, p(n) is True for all n>=0 Page 13

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COMP 170 L2 l Suppose b=0 l Intuitively n p(0) n p(0) => p(1) n p(0) /\ p(1) => p(2) (don’t have p(1) => p(2)) n p(0) /\ p(1) /\ p(2) => p(3), (don’t have p(2) => p(3)) n …. n So, p(n) is True for all n>=0 Page 14

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COMP 170 L2 Page 15

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COMP 170 L2 Strong Induction Implicitly Used in Proof of Euclid’s Division Theorem

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COMP 170 L2 Remarks Page 17

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COMP 170 L2 Summary Page 18

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