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MIT6_042JS10_lec08

# MIT6_042JS10_lec08 - The Idea of Induction Color the...

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lec 4M.1 Albert R Meyer, February 22, 2010 Induction lec 4M.2 Albert R Meyer, February 22, 2010 The Idea of Induction Color the integers 0 0 , 1 , 2 , 3 , 4 , 5 , … I tell you, 0 is red , & any int next to a red integer is red , then you know that all the ints are red ! lec 4M.3 Albert R Meyer, February 22, 2010 The Idea of Induction Color the integers 0 0 , 1 , 2 , 3 , 4 , 5 , … I tell you, 0 is red , & any int next to a red integer is red , then you know that all the ints are red ! lec 4M.4 Albert R Meyer, February 22, 2010 Induction Rule R ( 0 ), n . R ( n ) IMPLIES R ( n + 1 ) n . R ( n ) lec 4M.5 Albert R Meyer, February 22, 2010 Like Dominos… lec 4M.6 Albert R Meyer, February 22, 2010 Example Induction Proof Let’s prove: (for r 1) Image by MIT OpenCourseWare.

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lec 4M.7 Albert R Meyer, February 22, 2010 Statements in magenta form a template for inductive proofs: Proof: (by induction on n ) The induction hypothesis, P ( n ) , is: Example Induction Proof (for r 1) lec 4M.8 Albert R Meyer, February 22, 2010 Base Case ( n = 0 ) : Example Induction Proof 1 OK! lec 4M.10 Albert R Meyer, February 22, 2010 Inductive Step: Assume P ( n ) for some n 0 and prove P ( n+1 ) : Example Induction Proof lec 4M.11 Albert R Meyer, February 22, 2010 Now from induction hypothesis P ( n ) we have Example Induction Proof so add r n+1 to both sides lec 4M.12 Albert R Meyer, February 22, 2010 adding r n+1 to both sides, Example Induction Proof This proves P ( n+1 ) completing the proof by induction. lec 4M.15 Albert R Meyer, February 22, 2010 The MIT Stata Center Copyright © 2003, 2004, 2005 Norman Walsh. This work is licensed under a Creative Commons license.
2/22/10 3 lec 4M.16 Albert R Meyer, February 22, 2010 Design Mockup: Stata Lobby lec 4M.17 Albert R Meyer, February 22, 2010

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