week02 - Complete Mathematical Induction and Prime Numbers...

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Complete Mathematical Induction and Prime Numbers Original Notes adopted from September 18, 2001 (Week 2) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Complete Mathematical Induction If S N such that: a) 1 S b) (k +1 ) S whenever { 1,2,3, . .., k } S, then S = N. Example: What is Wrong with this Proof? Thm: For each n N, every set of people consists of people the same age. Proof: True for n = 1. Use Complete Mathematical Induction Assume true for n = 1,2,3,. ..k. Some k. Consider case n = k +1. Let S be any set of k +1 people, say S = { x 1 , x 2 ,... x k , x k+1 } Let S 0 = { x 1 , x 2 ,... x k } k people, so all some age by induction hypothesis. Let S 1 = { x 2 ,x 3 ,x 4 , . .. x k , x k+1 } k people, all same age. In particular, all same age as x 2 etc. for elements of S 0 . Everyone in S same age as x 2 By Mathematical Induction (Complete/Ordinary), every set of people consists of people the same age. Series with Positive Terms:
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This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto- Toronto.

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week02 - Complete Mathematical Induction and Prime Numbers...

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