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 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 ± 1 n 1 . Since n 1 ² n 0 then by P n 1 ± is true. Therefore, by inductive hypothesis P n 1 1 ± must be true. It follows then that P n 0 ± is true.
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This note was uploaded on 01/10/2011 for the course EE 100 taught by Professor Razasuleman during the Spring '10 term at University of Engineering & Technology.

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

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