induction-equivalents

# induction-equivalents - Forms of mathematical induction...

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Forms of mathematical induction March 19, 2010 We have covered three forms of mathematical induction: MI: ( P (0) ∧ ∀ ( P ( n ) = P ( n + 1))) = ⇒ ∀ nP ( n ). SMI: n (( k < n P ( k )) = P ( n )) = ⇒ ∀ nP ( n ). WO: S N ( S 6 = = ⇒ ∃ n S k < n k / S ). Theorem 1. The principles MI , SMI and WO are all equivalent. Proof. We will prove that SMI = MI = WO = SMI. Claim 1. (SMI = MI): We assume SMI, and we assume the hypoth- esis of MI, that is, P (0) ∧ ∀ n ( P ( n ) = P ( n + 1)) . We want to prove the conclusion of MI, that is, nP ( n ). To use SMI we need to prove n (( k < n P ( k )) = P ( n )). Let n N be arbitrary and assume the induction hypothesis k < n P ( k ). We have two cases: Case 1. ( n = 0 ): Then P ( n ) hold by our assumption of the hypothesis of MI. Case 2. ( n > 0 ): By our assumption of the hypothesis of MI, we know n ( P ( n ) = P ( n + 1)). In particular this holds with

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## This note was uploaded on 12/13/2011 for the course MHF 4102 taught by Professor Mitchelle during the Fall '11 term at University of Florida.

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induction-equivalents - Forms of mathematical induction...

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