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Strong Induction and Well-Ordering

CSE 2813 Discrete Structures. Mathematical Induction (Recap). A proof by mathematical induction that P(n) is true for every positive integer n consists of two steps:Basis step: The proposition P(1) is shown to be true.Inductive step: The implication P(k)?P(k 1) is shown to be true for every pos

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Strong Induction and Well-Ordering

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    1. CSE 2813 Discrete Structures Strong Induction and Well-Ordering Section 4.2

    2. CSE 2813 Discrete Structures Mathematical Induction (Recap) A proof by mathematical induction that P(n) is true for every positive integer n consists of two steps: Basis step: The proposition P(1) is shown to be true. Inductive step: The implication P(k)?P(k+1) is shown to be true for every positive integer k.

    3. CSE 2813 Discrete Structures Examples Use mathematical induction to prove that n < 2n for all positive integers n. Use mathematical induction to prove that n3n is divisible by 3 whenever n is a positive integer.

    4. CSE 2813 Discrete Structures Well-Ordered Sets The validity of mathematical induction follows from a fundamental axiom about the set of integers called the Well-Ordering Property Every nonempty set of nonnegative integers has a least element

    5. CSE 2813 Discrete Structures Why Mathematical Induction Is Valid Suppose we know that P(1) is true and that the proposition P(k)?P(k+1) is true for all positive integers k. To show that P(n) must be true for all positive integers, assume that there is at least one positive integer for which P(n) is false. Then the set S of positive integers for which P(n) is false is nonempty.

    6. CSE 2813 Discrete Structures Why Mathematical Induction Is Valid (Cont..) By the well-ordering property, S has a least element, which will be denoted by m. We know that m cannot be 1, since P(1) is true. Since m > 1, m-1 is a positive integer. Furthermore, since m-1 is less than m, it is not in S, so P(m-1) must be true.

    7. CSE 2813 Discrete Structures Why Mathematical Induction Is Valid (Cont..) Since the implication P(m-1) ? P(m) is also true, it must be the case that P(m) is true. This contradicts the choice of m. Hence, P(n) must be true for every positive integer n.

    8. CSE 2813 Discrete Structures Strong Induction To prove that P(n) is true for every positive integer n consists of two steps: Basis step: The proposition P(1) is shown to be true. Inductive step: The implication [P(1)?P(2)? ?P(k)]?P(k+1) is shown to be true for every positive integer k.

    9. CSE 2813 Discrete Structures Example Use Strong induction to show that if you can run one mile or two miles, and if you can always run two more miles once you have run a specified number of miles, then you can run any number of miles. Show that if n is an integer greater than 1, then n can be written as the product of primes.

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