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Instructor’s Resource Guide
to accompany
Discrete Mathematics
and Its Applications
8
th
Edition
Kenneth H. Rosen
Prepared by
Jerrold W. Grossman
and
Daniel R. Jordan
Copyright © 2019 The McGraw-Hill Companies, Inc. All rights reserved.
Preface
This Instructor’s Resource G uide for Discrete Mathematics and Its Applications, eighth edition, consists of five
items that an instructor of a course in discrete mathematics using the text should find useful.
• The bulk of this Guide consists of solutions to all the even-numbered exercises in the text, and thus com-
plements the Student’s Solutions Guide for Discrete Mathematics and Its Applications, eighth edition, which
contains solutions to the odd-numbered exercises. It is assumed that the user of the present manual has
access to that Guide as well. The solutions presented here are not necessarily the only ways of solving these
problems, of course, nor are the answers unique in all cases. These are complete solutions, although they are
somewhat less expository than the student-oriented solutions in the Student’s Solutions Guide.
• Several detailed course outlines are shown, for courses with different emphases and different student back-
grounds and ability levels. These suggested syllabi include courses with a mathematics emphasis, courses
with a computer science emphasis, one-term courses, and two-term courses.
• This Guide contains detailed teaching suggestions for instructors. There are chapter overviews, followed by
remarks on each section. Goals and prerequisites are stated, advice on teaching the section is offered, and
comments on the exercise sets are presented.
• Two sample tests are provided for each chapter and two sample final examinations—the first easier than the
second in each case. Solutions for the test questions are included. Instructors can draw on these sample tests
when constructing tests for their own classes, or they can provide them to students as samples with which
to prepare for the actual exams.
• Finally, this Guide contains a test bank of more than a thousand exam questions. Answers are included.
In addition to this Guide, you will find the companion website created for Discrete Mathematics and Its Appli-
cations an invaluable resource. Included here are a Web Resources Guide with links to external websites keyed
to the textbook, numerous Extra Examples to reinforce important topics, Interactive Demonstration Applets for
exploring key algorithms, Self Assessment question banks to gauge student understanding of core concepts, and
many helpful resources instructors can use for in-class teaching and homework assignments. See the section titled
“The Companion Website” on page xvi of the textbook for more details. The address is www.mhhe.com/rosen.
I want to thank Jerry Grossman for his extensive advice and assistance in the preparation of this Guide, Paul
Lorczak, Georgia Mederer, Lyndon Weberg, and Suzanne Zeitman for double-checking the solutions, and students
at Monmouth College and Oakland University for their input on preliminary versions of solutions to the exercises.
The test bank was produced by John Michaels, for whose excellent work I am most grateful. Some questions and
answers for this bank were contributed by Tao Jiang, Nancy Kinnersley, Antonette Logar, Thomas Roe, Zoltan
Szekely, and Bharti Temkin, to whom I also extend my appreciation.
It is possible that there are a few errors here, despite our best efforts at eliminating them. I would appreciate
hearing about all that you find, be they typographical or mathematical. Any other comments that will improve
subsequent editions of this book are always greatly appreciated. You can reach me using the Reporting of Errata
link on the companion website at www.mhhe.com/rosen.
Kenneth H. Rosen
i
Contents
Preface iii
Solutions for Even-numbered Exercises:
CHAPTER 1 The Foundations: Logic and Proofs 1
1.1 Propositional Logic 1
1.2 Applications of Propositional Logic 6
1.3 Propositional Equivalences 9
1.4 Predicates and Quantifiers 14
1.5 Nested Quantifiers 20
1.6 Rules of Inference 25
1.7 Introduction to Proofs 28
1.8 Proof Methods and Strategy 31
Supplementary Exercises for Chapter 1 34
CHAPTER 2 Basic Structures: Sets, Functions,
Sequences, Sums, and Matrices 38
2.1 Sets 38
2.2 Set Operations 41
2.3 Functions 45
2.4 Sequences and Summations 53
2.5 Cardinality of Sets 58
2.6 Matrices 61
Supplementary Exercises for Chapter 2 63
CHAPTER 3 Algorithms 66
3.1 Algorithms 66
3.2 The Growth of Functions 73
3.3 Complexity of Algorithms 78
Supplementary Exercises for Chapter 3 82
CHAPTER 4 Number Theory and Cryptography 87
4.1 Divisibility and Modular Arithmetic 87
4.2 Integer Representations and Algorithms 90
4.3 Primes and Greatest Common Divisors 94
4.4 Solving Congruences 101
4.5 Applications of Congruences 107
4.6 Cryptography 109
Supplementary Exercises for Chapter 4 110
v
CHAPTER 5 Induction and Recursion 115
5.1 Mathematical Induction 115
5.2 Strong Induction and Well-Ordering 124
5.3 Recursive Definitions and Structural Induction 130
5.4 Recursive Algorithms 136
5.5 Program Correctness 140
Supplementary Exercises for Chapter 5 141
CHAPTER 6 Counting 152
6.1 The Basics of Counting 152
6.2 The Pigeonhole Principle 158
6.3 Permutations and Combinations 161
6.4 Binomial Coefficients and Identities 164
6.5 Generalized Permutations and Combinations 167
6.6 Generating Permutations and Combinations 172
Supplementary Exercises for Chapter 6 173
CHAPTER 7 Discrete Probability 178
7.1 An Introduction to Discrete Probability 178
7.2 Probability Theory 180
7.3 Bayes’ Theorem 185
7.4 Expected Value and Variance 187
Supplementary Exercises for Chapter 7 192
CHAPTER 8 Advanced Counting Techniques 196
8.1 Applications of Recurrence Relations 196
8.2 Solving Linear Recurrence Relations 203
8.3 Divide-and-Conquer Algorithms
and Recurrence Relations 209
8.4 Generating Functions 213
8.5 Inclusion–Exclusion 224
8.6 Applications of Inclusion–Exclusion 226
Supplementary Exercises for Chapter 8 227
CHAPTER 9 Relations 231
9.1 Relations and Their Prop erties 231
9.2 n-ary Relations and Their Applications 236
9.3 Representing Relations 238
9.4 Closures of Relations 241
9.5 Equivalence Relations 244
9.6 Partial Orderings 249
Supplementary Exercises for Chapter 9 254
vi
CHAPTER 10 Graphs 258
10.1 Graphs and Graph Mo dels 258
10.2 Graph Terminology and Special Types of Graphs 260
10.3 Representing Graphs and Graph Isomorphism 265
10.4 Connectivity 270
10.5 Euler and Hamilton Paths 276
10.6 Shortest-Path Problems 280
10.7 Planar Graphs 282
10.8 Graph Coloring 284
Supplementary Exercises for Chapter 10 288
CHAPTER 11 Trees 294
11.1 Introduction to Trees 294
11.2 Applications of Trees 297
11.3 Tree Traversal 303
11.4 Spanning Trees 307
11.5 Minimum Spanning Trees 312
Supplementary Exercises for Chapter 11 314
CHAPTER 12 Boolean Algebra 318
12.1 Boolean Functions 318
12.2 Representing Boolean Functions 321
12.3 Logic Gates 323
12.4 Minimization of Circuits 325
Supplementary Exercises for Chapter 12 331
CHAPTER 13 Modeling Computation 334
13.1 Languages and Grammars 334
13.2 Finite-State Machines with Output 338
13.3 Finite-State Machines with No Output 341
13.4 Language Recognition 346
13.5 Turing Machines 349
Supplementary Exercises for Chapter 13 352
APPENDIXES 355
Appendix 1 Axioms for the Real Numbers
and the Positive Integers 355
Appendix 2 Exponential and Logarithmic Functions 356
Appendix 3 Pseudoco de 357
Suggested Syllabi 358
Teaching Suggestions 362
Chapter 1 362
Chapter 2 367
vii
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