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induction_1_print

# induction_1_print - V Adamchik 21-127 Concepts of...

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Mathematical Induction Victor Adamchik Fall of 2005 Lecture 1 (out of three) ° Plan 1. The Principle of Mathematical Induction 2. Induction Examples ° The Principle of Mathematical Induction Suppose we have some statement P ° n ± and we want to demonstrate that P ° n ± is true for all n ° ° . Even if we can provide proofs for P ° 0 ± , P ° 1 ± , ..., P ° k ± , where k is some large number, we have accomplished very little. However, there is a general method, the Princi- ple of Mathematical Induction . Induction is a defining difference between discrete and continuous mathematics. Principle of Induction . In order to show that ± n , P ° n ± holds, it suffices to establish the following two properties: (I1) Base case : Show that P ° 0 ± holds. (I2) Induction step : Assume that P ° n ± holds, and show that P ° n ² 1 ± also holds. In the induction step, the assumption that P ° n ± holds is called the Induction Hypothesis (IH). In more formal notation, this proof technique can be stated as ² P ° 0 ± ³ ± k ° P ° k ± ³ P ° k ² 1 ±±´ ´± n P ° n ± V. Adamchik 21-127: Concepts of Mathematics

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You can think of the proof by (mathematical) induction as a kind of recursive proof: Instead of attacking the problem directly, we only explain how to get a proof for P ° n ² 1 ± out of a proof for P ° n ± . How would you prove that the proof by induction indeed works?? Proof (by contradiction) Assume that for some values of n , P ° n ± is false. Let n 0 be the least such n that P ° n 0 ± is false. n 0 cannot be 0, because P ° 0 ± is true. Thus, n 0 must be in the form n 0 µ 1 ² n 1 . Since n 1 n 0 then by P ° n 1 ± is true. Therefore, by inductive hypothesis P ° n 1 ² 1 ± must be true. It follows then that P ° n 0 ± is true.
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induction_1_print - V Adamchik 21-127 Concepts of...

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