Chapter_04-4up - Sequences Sequences Induction Sums...

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Comp Sci 360 - Fall 2009 Induction Defnitions and ProoFs Sequences Sums Products John Hannan - Comp Sci 360 - Fall 2009 Sequences John Hannan - Comp Sci 360 - Fall 2009 Given: Set S oF elements & some property (predicate) P Goal: Prove that P holds For all elements oF S: ! x " S.P(x) Techniques? John Hannan - Comp Sci 360 - Fall 2009 Observing a Set 0 7 8 167 168
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John Hannan - Comp Sci 360 - Fall 2009 Observations? 2 3 4 6 9 8 10 12 14 15 16 18 20 21 John Hannan - Comp Sci 360 - Fall 2009 Observations? 4 2 6 8 10 12 14 16 3 6 9 12 15 18 Sets of Numbers Natural Numbers Real Numbers? John Hannan - Comp Sci 360 - Fall 2009 Inductive Definitions John Hannan - Comp Sci 360 - Fall 2009 Mathematical Induction To prove ! x " N.P(x) holds Inductive DeFnition of N: 1. 0 " N 2. if n " N then n+1 " N 1. Show P(0) holds 2. Show that if P(n) holds then P(n+1) must also hold
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John Hannan - Comp Sci 360 - Fall 2009 Examples ! i = n(n+1)/2 i=0 n 3 2n -1 is divisible by 8 A set with n elements has 2
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This note was uploaded on 10/04/2009 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Pennsylvania State University, University Park.

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Chapter_04-4up - Sequences Sequences Induction Sums...

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