4. Induction - MATH 224 Discrete Mathematics Induction...

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Unformatted text preview: MATH 224 Discrete Mathematics Induction Induction allows us to prove properties that hold over the integers are some infinite subset of the integers. So for example it provides for a way to prove that X3 X is a multiple of 3 for all values of X 0. In this case, there is also a simple proof that doesn't require induction. Induction can be illustrate by the problem of climbing a ladder. In order to climb a latter, you must be able to get to the first step and then from any step on the ladder to the next. The basis step requires that a property be true for the smallest value. Basis: Can you reach the first step? Yes Basis: Can you reach the first step? No 8/27/09 11 MATH 224 Discrete Mathematics Induction The inductive step is the heart of an induction proof. Here you must show that from any step on the ladder you can reach the next step. The inductive hypothesis states in this case that you are on an arbitrary step of the ladder. In your proofs you will be expected to explicitly state the, basis, inductive hypothesis and the inductive step. Inductive Inductive Hypothesis: States Hypothesis: States you are on a step of you are on a step of the ladder. the ladder. Inductive Step: Can you always reach the next step? Yes Inductive Step: Can you always reach the next step? No 8/27/09 22 ...
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4. Induction - MATH 224 Discrete Mathematics Induction...

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