dm-lec7-StrongFormofInduction-s - Discrete Mathematics...

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Discrete Mathematics CS204 Lecture #6 September 16, 2009 at KAIST 1
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Review Recall that - Mathematical induction, as a method of proof, is based on the following property of the set of positive integers. For n Z + , S (1) ( n S ( n ) S ( n + 1)) → ∀ n S ( n ) where S ( n ) is a propositional function defined on the set of positive integers. - The implication n S ( n ) S ( n + 1) is supposedly easier to prove than n S ( n ) . 2
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Strong Form of Induction - The strong form of mathematical induction allows us to assume the truth of all of the preceding statements. - At the inductive step of strong form of induction, we prove the implication ( S ( k ) for all k with 1 k < n ) S ( n ) which is a weaker statement than n S ( n - 1) S ( n ) . Note that if S ( n - 1) S ( n ) is true, then ( S ( k ) for all k with 1 k < n ) S ( n ) is true. - The strong form of induction obtains the same conclusion from a priori weaker statements. 3
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Strong Form of Mathematical Induction Suppose that we have a propositional function S ( n )
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