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Unformatted text preview: 1) P (0) 2) ∀ k ∈ N , ( ∀ m ≤ k , P ( m )) → P ( k +1) Inductive hypothesis Strong inductive hypothesis Example Claim : Every postage value n ≥ 12 cents can be made up of 4cent and 5cent stamps — Mathematically, ∀ n ≥ 12, ∃ a ∈ N , ∃ b ∈ N , n = 4 a + 5 b Proof by induction on n : Base cases ( n = 12, 13, 14, 15): Since 12 = 4 × 3+5 × 0, 13 = 4 × 2+5 × 1, 14 = 4 × 1+5 × 2, 15 = 4 × 0+5 × 3, the base cases all hold. Inductive step: Let k ≥ 15 such that ∀ m ≤ k and m ≥ 12, ∃ a ∈ N , ∃ b ∈ N , m = 4 a + 5 b [strong IH] We need to show that ∃ p ∈ N , ∃ q ∈ N , k +1 = 4 p + 5 q Now k +1 = ( k − 3)+4. Since k ≥ 15, k − 3 ≥ 12 and hence by the strong IH, ∃ a ∈ N , ∃ b ∈ N , k − 3 = 4 a + 5 b Hence k +1 = 4 a + 5 b + 4 = 4( a +1) + 5 b . Hence the statement holds for k +1, which completes the proof by (strong) induction....
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This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.
 Spring '10
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