Discrete Mathematics Demystified BGU Math Homepage

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Discrete Mathematics,Demystified,Demystified Series. Accounting Demysti ed JavaScript Demysti ed, Advanced Calculus Demysti ed Lean Six Sigma Demysti ed. Advanced Physics Demysti ed Linear Algebra Demysti ed. Advanced Statistics Demysti ed Macroeconomics Demysti ed. Algebra Demysti ed Management Accounting Demysti ed. Alternative Energy Demysti ed Math Proofs Demysti ed. Anatomy Demysti ed Math Word Problems Demysti ed,Astronomy Demysti ed Mathematica Demysti ed. Audio Demysti ed MATLAB Demysti ed, Biochemistry Demysti ed Medical Billing and Coding Demysti ed. Biology Demysti ed Medical Charting Demysti ed, Biotechnology Demysti ed Medical Surgical Nursing Demysti ed.
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Hedge Funds Demysti ed Technical Analysis Demysti ed. Investing Demysti ed Technical Math Demysti ed,Italian Demysti ed Trigonometry Demysti ed. Java Demysti ed Vitamins and Minerals Demysti ed,Discrete Mathematics. Demystified,Steven G Krantz,New York Chicago San Francisco Lisbon London. Madrid Mexico City Milan New Delhi San Juan,Seoul Singapore Sydney Toronto. Copyright 2009 by The McGraw Hill Companies Inc All rights reserved Except as permitted under the United States Copyright. Act of 1976 no part of this publication may be reproduced or distributed in any form or by any means or stored in a database or. retrieval system without the prior written permission of the publisher. ISBN 978 0 07 154949 3,MHID 0 07 154949 8, The material in this eBook also appears in the print version of this title ISBN 978 0 07 154948 6 MHID 0 07 154948 X.
All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a. trademarked name we use names in an editorial fashion only and to the benefit of the trademark owner with no intention of. infringement of the trademark Where such designations appear in this book they have been printed with initial caps. McGraw Hill eBooks are available at special quantity discounts to use as premiums and sales promotions or for use in corporate. training programs To contact a representative please visit the Contact Us page at www mhprofessional com. TERMS OF USE, This is a copyrighted work and The McGraw Hill Companies Inc McGraw Hill and its licensors reserve all rights in and to. the work Use of this work is subject to these terms Except as permitted under the Copyright Act of 1976 and the right to store and. retrieve one copy of the work you may not decompile disassemble reverse engineer reproduce modify create derivative works. based upon transmit distribute disseminate sell publish or sublicense the work or any part of it without McGraw Hill s prior con. sent You may use the work for your own noncommercial and personal use any other use of the work is strictly prohibited Your. right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED AS IS McGRAW HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES. AS TO THE ACCURACY ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE. WORK INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR. OTHERWISE AND EXPRESSLY DISCLAIM ANY WARRANTY EXPRESS OR IMPLIED INCLUDING BUT NOT LIMIT. ED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw Hill and. its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation. will be uninterrupted or error free Neither McGraw Hill nor its licensors shall be liable to you or anyone else for any inaccuracy. error or omission regardless of cause in the work or for any damages resulting therefrom McGraw Hill has no responsibility for. the content of any information accessed through the work Under no circumstances shall McGraw Hill and or its licensors be liable. for any indirect incidental special punitive consequential or similar damages that result from the use of or inability to use the. work even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim. or cause whatsoever whether such claim or cause arises in contract tort or otherwise. To the memory of J W T Youngs,ABOUT THE AUTHOR, Steven G Krantz Ph D is a professor of mathematics at Washington University. in St Louis Missouri He just nished a stint as deputy director at the American. Institute of Mathematics Dr Krantz is an award winning teacher and the author. of How to Teach Mathematics Calculus Demysti ed and Differential Equations. Demysti ed among other books,Preface xiii,CHAPTER 1 Logic 1. 1 1 Sentential Logic 2,1 2 And and Or 3,1 4 If Then 8. 1 5 Contrapositive Converse and Iff 12,1 6 Quanti ers 16.
Exercises 20,CHAPTER 2 Methods of Mathematical Proof 23. 2 1 What Is a Proof 23,2 2 Direct Proof 24,2 3 Proof by Contradiction 29. 2 4 Proof by Induction 32,2 5 Other Methods of Proof 37. Exercises 40,CHAPTER 3 Set Theory 41,3 1 Rudiments 41. 3 2 Elements of Set Theory 42,3 3 Venn Diagrams 46.
viii Discrete Mathematics Demystified,3 4 Further Ideas in Elementary Set Theory 47. Exercises 49,CHAPTER 4 Functions and Relations 51,4 1 A Word About Number Systems 51. 4 2 Relations and Functions 53,4 3 Functions 56,4 4 Combining Functions 59. 4 5 Types of Functions 63,Exercises 65,CHAPTER 5 Number Systems 67. 5 1 Preliminary Remarks 67,5 2 The Natural Number System 68.
5 3 The Integers 73,5 4 The Rational Numbers 79,5 5 The Real Number System 86. 5 6 The Nonstandard Real Number System 94,5 7 The Complex Numbers 96. 5 8 The Quaternions the Cayley Numbers,and Beyond 101. Exercises 102,CHAPTER 6 Counting Arguments 105,6 1 The Pigeonhole Principle 105. 6 2 Orders and Permutations 108,6 3 Choosing and the Binomial Coef cients 110.
6 4 Other Counting Arguments 113,6 5 Generating Functions 118. 6 6 A Few Words About Recursion Relations 121,6 7 Probability 124. 6 8 Pascal s Triangle 127,6 9 Ramsey Theory 130,Exercises 132. Contents ix,CHAPTER 7 Matrices 135,7 1 What Is a Matrix 135. 7 2 Fundamental Operations on Matrices 136,7 3 Gaussian Elimination 139.
7 4 The Inverse of a Matrix 145,7 5 Markov Chains 153. 7 6 Linear Programming 156,Exercises 161,CHAPTER 8 Graph Theory 163. 8 1 Introduction 163,8 2 Fundamental Ideas of Graph Theory 165. 8 3 Application to the Ko nigsberg,Bridge Problem 169. 8 4 Coloring Problems 172,8 5 The Traveling Salesman Problem 178.
Exercises 181,CHAPTER 9 Number Theory 183,9 1 Divisibility 183. 9 2 Primes 185,9 3 Modular Arithmetic 186,9 4 The Concept of a Group 187. 9 5 Some Theorems of Fermat 196,Exercises 197,CHAPTER 10 Cryptography 199. 10 1 Background on Alan Turing 199,10 2 The Turing Machine 200. 10 3 More on the Life of Alan Turing 202,10 4 What Is Cryptography 203.
10 5 Encryption by Way of Af ne Transformations 209. 10 6 Digraph Transformations 216,x Discrete Mathematics Demystified. 10 7 RSA Encryption 221,Exercises 233,CHAPTER 11 Boolean Algebra 235. 11 1 Description of Boolean Algebra 235,11 2 Axioms of Boolean Algebra 236. 11 3 Theorems in Boolean Algebra 238,11 4 Illustration of the Use of Boolean Logic 239. Exercises 241,CHAPTER 12 Sequences 243,12 1 Introductory Remarks 243.
12 2 In nite Sequences of Real Numbers 244,12 3 The Tail of a Sequence 250. 12 4 A Basic Theorem 250,12 5 The Pinching Theorem 253. 12 6 Some Special Sequences 254,Exercises 256,CHAPTER 13 Series 257. 13 1 Fundamental Ideas 257,13 2 Some Examples 260,13 3 The Harmonic Series 263. 13 4 Series of Powers 265,13 5 Repeating Decimals 266.
13 6 An Application 268,13 7 A Basic Test for Convergence 269. 13 8 Basic Properties of Series 270,13 9 Geometric Series 273. 13 10 Convergence of p Series 279,13 11 The Comparison Test 283. 13 12 A Test for Divergence 288,13 13 The Ratio Test 291. 13 14 The Root Test 294,Exercises 298,Contents xi,Final Exam 301.
Solutions to Exercises 325,Bibliography 347,This page intentionally left blank. In today s world analytical thinking is a critical part of any solid education An im. portant segment of this kind of reasoning one that cuts across many disciplines is. discrete mathematics Discrete math concerns counting probability sophisticated. forms of addition and limit processes over discrete sets Combinatorics graph. theory the idea of function recurrence relations permutations and set theory are. all part of discrete math Sequences and series are among the most important ap. plications of these ideas, Discrete mathematics is an essential part of the foundations of theoretical. computer science statistics probability theory and algebra The ideas come up. repeatedly in different parts of calculus Many would argue that discrete math is. the most important component of all modern mathematical thought. Most basic math courses at the freshman and sophomore level are oriented. toward problem solving Students can rely heavily on the provided examples as a. crutch to learn the basic techniques and pass the exams Discrete mathematics is by. contrast rather theoretical It involves proofs and ideas and abstraction Freshman. and sophomores in college these days have little experience with theory or with. abstract thinking They simply are not intellectually prepared for such material. Steven G Krantz is an award winning teacher author of the book How to Teach. Mathematics He knows how to present mathematical ideas in a concrete fashion. that students can absorb and master in a comfortable fashion He can explain even. abstract concepts in a hands on fashion making the learning process natural and. uid Examples can be made tactile and real thus helping students to nesse abstract. technicalities This book will serve as an ideal supplement to any standard text It. will help students over the traditional hump that the rst theoretical math course. constitutes It will make the course palatable Krantz has already authored two. successful Demysti ed books, The good news is that discrete math particularly sequences and series can. be illustrated with concrete examples from the real world They can be made to. be realistic and approachable Thus the rather dif cult set of ideas can be made. accessible to a broad audience of students For today s audience consisting not. xiv Discrete Mathematics Demystified, only of mathematics students but of engineers physicists premedical students. social scientists and others this feature is especially important. A typical audience for this book will be freshman and sophomore students in the. mathematical sciences in engineering in physics and in any eld where analytical. thinking will play a role Today premedical students nursing students business. students and many others take some version of calculus or discrete math or both. They will de nitely need help with these theoretical topics. This text has several key features that make it unique and useful. 1 The book makes abstract ideas concrete All concepts are presented succinctly. and clearly, 2 Real world examples illustrate ideas and make them accessible.
3 Applications and examples come from real believable contexts that are. familiar and meaningful, 4 Exercises develop both routine and analytical thinking skills. 5 The book relates discrete math ideas to other parts of mathematics and. Discrete Mathematics Demysti ed explains this panorama of ideas in a step by. step and accessible manner The author a renowned teacher and expositor has a. strong sense of the level of the students who will read this book their backgrounds. and their strengths and can present the material in accessible morsels that the student. can study on his or her own Well chosen examples and cognate exercises will. viii Discrete Mathematics Demystified 3 4 Further Ideas in Elementary Set Theory 47 Exercises 49 CHAPTER 4 Functions and Relations 51 4 1 A Word About Number Systems 51 4 2 Relations and Functions 53 4 3 Functions 56 4 4 Combining Functions 59 4 5 Types of Functions 63 Exercises 65 CHAPTER 5 Number Systems 67 5 1 Preliminary Remarks 67

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