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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 '07
 SELMAN
 Mathematical Induction, Natural number, Prime number, Strong Induction, induction hypothesis

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