StrongInduction.pdf - Section 4.4 Summary Strong Induction Example Proofs using Strong Induction 1 Strong Induction Strong Induction To prove that P(n

StrongInduction.pdf - Section 4.4 Summary Strong Induction...

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10/7/2019 1 Section 4.4 Summary Strong Induction Example Proofs using Strong Induction
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10/7/2019 2 Strong Induction Strong Induction : To prove that P ( n ) is true for all positive integers n , where P ( n ) is a propositional function, complete two steps: Basis Step : Verify that the proposition P ( 1 ) is true. Inductive Step : Show the conditional statement [ P ( 1 ) P ( 2 ) ∧∙∙∙ ∧ P ( k )] P ( k + 1 ) holds for all positive integers k . Strong Induction is sometimes called the second principle of mathematical induction or complete induction . Strong Induction and the Infinite Ladder Strong induction tells us that we can reach all rungs if: 1. We can reach the first rung of the ladder. 2. For every integer k , if we can reach the first k rungs, then we can reach the ( k + 1 )st rung. To conclude that we can reach every rung by strong induction: BASIS STEP: P ( 1 ) holds INDUCTIVE STEP: Assume P ( 1 ) P ( 2 ) ∧∙∙∙ ∧ P ( k ) holds for an arbitrary integer k , and show that P ( k + 1 ) must also hold .
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