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induction

# induction - CS 2800 Discrete Structures(Fall 11 Oct.26 2011...

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CS 2800: Discrete Structures (Fall ’11) Oct.26, 2011 Induction Prepared by Doo San Baik(db478) Concept of Inductive Proof When you think of induction, one of the best analogies to think about is ladder . When you climb up the ladder, you have to step on the lower step and need to go up based on it. After we climb up the several steps, we can go up further by assuming that the step you are stepping on exists. With the terms we have covered in class we can make such analogies. 1. Base Case : The first step in the ladder you are stepping on 2. Induction Hypothesis : The steps you are assuming to exist Weak Induction : The step that you are currently stepping on Strong Induction : The steps that you have stepped on before including the current one 3. Inductive Step : Going up further based on the steps we assumed to exist Components of Inductive Proof Inductive proof is composed of 3 major parts : Base Case, Induction Hypothesis, Inductive Step. When you write down the solutions using induction, it is always a great idea to think about this template. 1. Base Case : One or more particular cases that represent the most basic case. (e.g. n=1 to prove a statement in the range of positive integer) 2. Induction Hypothesis : Assumption that we would like to be based on. (e.g. Let’s assume that P(k) holds) 3. Inductive Step : Prove the next step based on the induction hypothesis. (i.e. Show that Induction hypothesis P(k) implies P(k+1)) Weak Induction, Strong Induction This part was not covered in the lecture explicitly. However, it is always a good idea to keep this in mind regarding the differences between weak induction and strong induction.

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induction - CS 2800 Discrete Structures(Fall 11 Oct.26 2011...

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