tailieunhanh - Lecture Discrete mathematics and its applications - Lecture 8

This chapter include objectives: Identify at least three benefits of using a network, differentiate between LANs and WANs, identify at (east three common network topologies, name two common network media, identify network hardware and linking devices. | Induction and recursion Chapter 5 With Question/Answer Animations Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Mathematical Induction Strong Induction Well-Ordering Recursive Definitions Structural Induction Recursive Algorithms Program Correctness (not yet included in overheads) Mathematical Induction Section Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: We can reach the first rung of the ladder. If we can reach a particular rung of the ladder, then we can reach the next rung. From (1), we can reach the first rung. Then by applying (2), we can reach the second rung. Applying (2) again, the third rung. And so on. We can apply (2) any number of times to reach any particular rung, no matter how high up. This example motivates proof by mathematical induction. Principle of Mathematical Induction Principle of Mathematical Induction: To prove that P(n) is true for all positive integers n, we complete these steps: Basis Step: Show that P(1) is true. Inductive Step: Show that P(k) → P(k + 1) is true for all positive integers k. To complete the inductive step, assuming the inductive hypothesis that P(k) holds for an arbitrary integer k, show that must P(k + 1) be true. Climbing an Infinite Ladder Example: BASIS STEP: By (1), we can reach rung 1. INDUCTIVE STEP: Assume the inductive hypothesis that we can reach rung k. Then by (2), we can reach rung k + 1. Hence, P(k) → P(k + 1) is true for all positive integers k. We can reach every rung on the ladder. Important Points About Using Mathematical Induction Mathematical induction can be expressed as the rule of inference where the domain is the set of positive integers. In a proof by mathematical induction, we don’t . | Induction and recursion Chapter 5 With Question/Answer Animations Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Mathematical Induction Strong Induction Well-Ordering Recursive Definitions Structural Induction Recursive Algorithms Program Correctness (not yet included in overheads) Mathematical Induction Section Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: We can reach the first rung of the ladder. If we can reach a particular rung of the ladder, then we can reach the next rung. From (1), we can reach the first rung. Then by applying (2), we can reach the second rung. Applying (2) again, the third rung. And so on. We can apply (2) any number of times to reach any particular .