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21a-Mathematical-Induction

# 21a-Mathematical-Induction - Mathematical Introduction...

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Mathematical Induction II Discrete Mathematics

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Discrete Mathematics – Mathematical Induction II 21-2 Principle of Mathematical Induction Principle of mathematical induction : To prove that a statement that assert that some property P(n) is true for all positive integers n, we complete two steps Basis step : We verify that P(1) is true. Inductive step : We show that the conditional statement P(k) P(k + 1) is true for all positive integers k To prove the conditional statement, we assume that P(k) is true (it is called inductive hypothesis ) and show that under this assumption P(k + 1) is also true
Discrete Mathematics – Mathematical Induction II 21-3 Analysis of Algorithms Consider the following problem There is a group of proposed talks to be given. We want to schedule as many talks as possible in the main lecture room. Let be the talks, talk begins at time and ends at time . (No two lectures can proceed at the same time, but a lecture can begin at the same time another one ends.) We assume that . 9:00 10:00 11:00 12:00

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Greedy algorithm: At every step choose a talk with the earliest ending time among all those talks that begin after all talks already scheduled end. We prove that the greedy algorithm is optimal in the sense that it always schedules the most talks possible in the main lecture hall. 9:00 10:00 11:00 12:00 Discrete Mathematics – Mathematical Induction II 21-4 Greedy Algorithm
Discrete Mathematics – Mathematical Induction II 21-5 Greedy Algorithm (cntd) Let P(n) be the proposition that if the greedy algorithm schedules n talks, then it is not possible to schedule more than n talks. Basis step. Suppose that the greedy algorithm has scheduled only one talk, . This means that every other talk starts before , and ends after . Hence, at time each of the remaining talks needs to use the lecture hall. No two talks can be scheduled because of that. This proves P(1). Inductive step. Suppose that P(k) is true, that is, if the greedy algorithm schedules k talks, it is not possible to schedule more than k talks. We prove P(k + 1), that is, if the algorithm schedules k + 1 talks then this is the optimal number.

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Discrete Mathematics – Mathematical Induction II 21-6 Greedy Algorithm (cntd) Suppose that the algorithm has selected k + 1 talks.
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21a-Mathematical-Induction - Mathematical Introduction...

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