ics141-lecture19-Induction

ics141-lecture19-Induction - 19-1ICS 141: Discrete...

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Unformatted text preview: 19-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga19-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 19Chapter 4. Induction and Recursion4.1 Mathematical Induction4.2 Strong Induction and Well-Ordering4.3 Recursive Definitions and Structural Induction19-3ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of Discrete Mathematics and Its Applications written by Kenneth H. Rosen.Some slides were done by Prof. Baek19-4ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiMathematical InductionA powerful, rigorous technique for proving that a predicate P(n) is true for everypositive integers n, no matter how large.Essentially a domino effect principle.Based on a predicate-logic inference rule:P(1)2200k1[P(k)P(k+1)]2200n1P(n)The First Principleof MathematicalInduction19-5ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiThe Domino EffectPremise #1:Domino #1 falls.Premise #2:For every kZ+, if domino #kfalls, then so does domino #k+1.Conclusion:All of the dominoes fall down!k1k + 1k1 2 3 4 5Note:this works even if there are infinitely many dominoes!19-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiMathematical Induction Recap.PRINCIPLE OF MATHEMATICAL INDUCTION:To prove that a predicate P(n) is true for all positive integers n, we complete two steps:BASIS STEP: Verify that P(1)is trueINDUCTIVE STEP: Show that the conditional statement P(k)P(k+1) is true for all positive integers kInductive Hypothesis19-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiInduction Example (1st princ.)Example 2: Conjecture a formula for the sum of the first npositive odd integers. Then prove your conjecture using mathematical induction.Practical Method for General Problem Solving.Special Case: Deriving a FormulaStep 1. Calculate the result for some small casesStep 2. Guess a formula to match all those casesStep 3. Verify your guess in the general case19-8ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiExample cont....
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.

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ics141-lecture19-Induction - 19-1ICS 141: Discrete...

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