MIT6_042JS10_lec08

MIT6_042JS10_lec08 - The Idea of Induction Color the...

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±²±±²³´µ ³µ 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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±²±±²³´µ ±µ 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.
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MIT6_042JS10_lec08 - The Idea of Induction Color the...

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