DS_Lecture_21.pdf - Lecture 21 Strong Induction and Well-Ordering Dr Chengjiang Long Computer Vision Researcher at Kitware Inc Adjunct Professor at SUNY

DS_Lecture_21.pdf - Lecture 21 Strong Induction and...

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Lecture 21: Strong Induction and Well-Ordering Dr. Chengjiang Long Computer Vision Researcher at Kitware Inc. Adjunct Professor at SUNY at Albany. Email: [email protected]
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C. Long Lecture 20 October 24, 2018 2 ICEN/ICSI210 Discrete Structures Outline Strong Induction Well-Ordering Principle
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C. Long Lecture 20 October 24, 2018 3 ICEN/ICSI210 Discrete Structures Outline Strong Induction Well-Ordering Principle
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C. Long Lecture 20 October 24, 2018 4 ICEN/ICSI210 Discrete Structures Proofs Basic proof methods : Direct, Indirect, Contradiction, By Cases, Equivalences Proof of quantified statements : There exists x with some property P(x). – It is sufficient to find one element for which the property holds. For all x some property P(x) holds . – Proofs of ‘For all x some property P(x) holds’ must cover all x and can be harder. Mathematical induction is a technique that can be applied to prove the universal statements for sets of positive integers or their associated sequences.
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C. Long Lecture 20 October 24, 2018 5 ICEN/ICSI210 Discrete Structures Mathematical induction Used to prove statements of the form x P(x) where " ∈ $ % Mathematical induction proofs consists of two steps: 1) Basis : The proposition P(1) is true. 2) Inductive Step : The implication P(n) à P(n+1), is true for all positive n. Therefore we conclude x P(x). Based on the well-ordering property : Every nonempty set of nonnegative integers has a least element
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C. Long Lecture 20 October 24, 2018 6 ICEN/ICSI210 Discrete Structures Correctness of Mathematical Induction Suppose P(1) is true and P(n) à P(n+1) is true for all positive integers n. Want to show x P(x). Assume there is at least one n such that P(n) is false. Let S be the set of nonnegative integers where P(n) is false. Thus S ≠ ∅ . Well-Ordering Property : Every nonempty set of nonnegative integers has a least element. By the Well-Ordering Property , S has a least member, say k. k > 1, since P(1) is true. This implies k - 1 > 0 and P(k-1) is true (since k is the smallest integer where P(k) is false).
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