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lecture8_9 - Discrete Computation Structures CSE 2353 Bhanu...

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Discrete Computation Structures CSE 2353 Bhanu Kapoor, PhD Department of Computer Science & Engineering Southern Methodist University, Dallas, TX [email protected] , 214-336-4973 September 22&24, 2009 Embrey 129, SMU, Dallas, Texas Lecture 08, 09 1
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Lecture 08 and 09 Agenda ± Review Fermat’s Factoring Method ± Mathematical Induction ± Matrices 2
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Mathematical Induction ± Proof Techniques: Logic, Induction powerful rigorous technique for proving ± A powerful, rigorous technique for proving that a predicate P ( n ) is true for every natural number n , no matter how large. ± Based on a predicate-logic inference rule: ± P (0) n 0( P ( n ) P ( n +1)) ∴∀ n 0 P ( n ) 3
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Outline of an Inductive Proof ± Want to prove nP ( n ) … ase case r asis step Prove ) ± Base case (or basis step ): Prove P (0). ± Inductive step : Prove ( n ) P ( n +1). ² e.g. use a direct proof: ± Let n N , assume P ( n ). ( inductive hypothesis ) ± Under this assumption, prove P ( n +1). ± Inductive inference rule then gives n P ( n ). 4
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Induction Example ± Prove that n 1 0 (1 ) / ( 1 ) , 1 kn k rr r r + = = −− 5
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Another Induction Example ± Prove that , n <2 n . Let P ( n )=( n <2 n ) 0 n ² Base case: P (0)=(0<2 0 )=(0<1)= T . ² Inductive step: For prove 0 n P ( n ) P ( n +1). ± Assuming n <2 n , prove n +1 < 2 n +1 . ± Note n + 1 < 2 n + 1 (by inductive hypothesis) < 2 n + 2 n (because 1<2=2 2 0 2 2 n -1 = 2 n ) = 2 n +1 ± So n + 1 < 2 n +1 , and we’re done. 6
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Validity of Induction Prove: if n 0( P ( n ) P ( n +1)), then k 0 P ( k ) ) Given any 1)) implies (a) Given any k 0, n 0 ( P ( n ) P ( n +1)) implies ( P (0) P (1)) ( P (1) P (2)) ( P ( k 1) P ( k )) (b) Using hypothetical syllogism k -1 times we have P (0) P ( k ) (c) P (0) and modus ponens gives P ( k ).
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lecture8_9 - Discrete Computation Structures CSE 2353 Bhanu...

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