induction-equivalents

induction-equivalents - Forms of mathematical induction...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 2

induction-equivalents - Forms of mathematical induction...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online