lecture23 - Lecture 23 Induction Strong Induction Induction...

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Lecture 23 Induction, Strong Induction
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Induction A powerful technique to prove statements of the form n P(n) Examples: n, n < 2 n 2 ) 1 ( , 1 n n i n n i As usual, we need to specify the domain of n
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Strategy We will show two things: 1. Basis: P(1) is true 2. Inductive Step: k (P(k) From these two, we can infer that: n P(n) n = {1,2,3,…} So, let’s start. . Well Ordering Property: Every non-empty set of positive integers has a least element
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Generalized DeMorgan for Sets n i i n i i A A 1 1
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Harmonic Numbers i H i 1 ... 3 1 2 1 1 Prove that, for all n 0 2 1 2 n H n
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Strong Induction
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Recall the (regular) induction We will show two things: 1. Basis: P(1) is true 2. Inductive Step: k (P(k) P(k+1)) is true The Inductive step for strong induction: Inductive Step: k ( [P(1) P(2) P(k)] P(k+1) ) is true i.e. We assume that all of P(1), P(2), …, P(k) is true, and show that this implies that P(k+1) must be true.
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Prime Factorization (existence) Every prime p number can be written as a product of primes (using the book’s
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lecture23 - Lecture 23 Induction Strong Induction Induction...

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