Student's Solutions Guide
to accompany
··Discrete· .
.:..· ~' :
...·.Mathematics·
. ' .
and Its
Applications
SEVENTH EDITION
Prepared by
Jerrold Grossman .
, Student's Solutions Guide
to accompany
Discrete Mathematics
and Its Applications
Seventh Edition
Kenneth H. Rosen
Monmouth University
(and formerly AT&T Laboratories)
Prepared by
Jerrold W. Grossman
Oakland University
~~onnect
Learn
• Succeed"
, The McGrow·Hill Companies
~~onnect
learn
• Succeed'
Student's Solutions Guide to accompany
DISCRETE MATHEMATICS AND ITS APPLICATIONS, SEVENTH EDITION
KENNETH H. ROSEN
Published by McGraw-Hill Higher Education, an imprint of The McGraw-Hill Compames, Inc., 1221 Avenue of the Americas, New
York, NY 10020. Copyright© 2012 and 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Pnnted in the United States of America.
No part of this publication may be reproduced or distributed many form or by any means, or stored ma database or retrieval system,
without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, network or other electronic
storage or transm1ss10n, or broadcast for distance leammg.
1234567890QDB~DB10987654321
ISBN: 978-0-07-735350-6
MHID: 0-07-735350-1
www.mhhe.com
, Preface
This Student's Solutions Guide for Discrete Mathematics and Its Applications, seventh edition,
contains several useful and important study aids.
• SOLUTIONS TO ODD-NUMBERED EXERCISES
The bulk of this work consists of solutions to all the odd-numbered exercises in
the text. These are not just answers, but complete, worked-out solutions, showing how
the principles discussed in the text and examples are applied to the problems. I have also
added bits of wisdom, insights, warnings about errors to avoid, and extra comments that
go beyond the question as posed. Furthermore, at the beginning of each section you will
find some general words of advice and hints on approaching the exercises in that section.
• REFERENCES FOR CHAPTER REVIEWS
Exact page references, theorem and example references, and answers are provided as a
guide for all the chapter review questions in the text. This will make reviewing for
tests and quizzes particularly easy.
• A GUIDE TO WRITING PROOFS
Near the end of this book is a section on writing proofs, a skill that most stu-
dents find difficult to master. Proofs are introduced formally in Chapter 1 (and proofs by
mathematical induction are studied in Chapter 5), but exercises throughout the text ask
for proofs of propositions. Reading this section when studying Sections 1.6-1.8, and then
periodically thereafter, will be rewarded, as your proof-writing ability grows.
• REFERENCES AND ADVICE ON THE WRITING PROJECTS
Near the end of this book you will find some general advice on the Writing Projects
given at the end of each chapter. There is a discussion of various resources available in the
mathematical sciences (such as Mathematical Reviews and the World Wide Web), tips on
doing library research, and some suggestions on how to write well. In addition, there
is a rather extensive bibliography of books and articles that will be useful when
researching these projects. We also provide specific hints and suggestions for each project,
with pointers to the references; these can be found in the solutions section of this manual,
at the end of each chapter.
• SAMPLE CHAPTER TESTS
Near the end of this book you will find a sequence of 13 chapter tests, comparable
to what might be given in a course. You can take these sample tests in a simulated test
setting as practice for the real thing. Complete solutions are provided, of course.
• PROBLEM-SOLVING TIPS AND LIST OF COMMON MISTAKES
People beginning any endeavor tend to make the same kinds of mistakes. This is
especially true in the study of mathematics. I have included a detailed list of common
misconceptions that students of discrete mathematics often have and the kinds of errors
they tend to make. Specific examples are given. It will be useful for you to review this list
from time to time, to make sure that you are not falling into these common traps. Also
included in this section is general advice on solving problems in mathematics, which you
will find helpful as you tackle the exercises.
• CRIB SHEETS
Finally, I have prepared a set of 13 single-page "crib sheets," one for each chapter
of the book. They provide a quick summary of all the important concepts, definitions, and
lll
to accompany
··Discrete· .
.:..· ~' :
...·.Mathematics·
. ' .
and Its
Applications
SEVENTH EDITION
Prepared by
Jerrold Grossman .
, Student's Solutions Guide
to accompany
Discrete Mathematics
and Its Applications
Seventh Edition
Kenneth H. Rosen
Monmouth University
(and formerly AT&T Laboratories)
Prepared by
Jerrold W. Grossman
Oakland University
~~onnect
Learn
• Succeed"
, The McGrow·Hill Companies
~~onnect
learn
• Succeed'
Student's Solutions Guide to accompany
DISCRETE MATHEMATICS AND ITS APPLICATIONS, SEVENTH EDITION
KENNETH H. ROSEN
Published by McGraw-Hill Higher Education, an imprint of The McGraw-Hill Compames, Inc., 1221 Avenue of the Americas, New
York, NY 10020. Copyright© 2012 and 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Pnnted in the United States of America.
No part of this publication may be reproduced or distributed many form or by any means, or stored ma database or retrieval system,
without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, network or other electronic
storage or transm1ss10n, or broadcast for distance leammg.
1234567890QDB~DB10987654321
ISBN: 978-0-07-735350-6
MHID: 0-07-735350-1
www.mhhe.com
, Preface
This Student's Solutions Guide for Discrete Mathematics and Its Applications, seventh edition,
contains several useful and important study aids.
• SOLUTIONS TO ODD-NUMBERED EXERCISES
The bulk of this work consists of solutions to all the odd-numbered exercises in
the text. These are not just answers, but complete, worked-out solutions, showing how
the principles discussed in the text and examples are applied to the problems. I have also
added bits of wisdom, insights, warnings about errors to avoid, and extra comments that
go beyond the question as posed. Furthermore, at the beginning of each section you will
find some general words of advice and hints on approaching the exercises in that section.
• REFERENCES FOR CHAPTER REVIEWS
Exact page references, theorem and example references, and answers are provided as a
guide for all the chapter review questions in the text. This will make reviewing for
tests and quizzes particularly easy.
• A GUIDE TO WRITING PROOFS
Near the end of this book is a section on writing proofs, a skill that most stu-
dents find difficult to master. Proofs are introduced formally in Chapter 1 (and proofs by
mathematical induction are studied in Chapter 5), but exercises throughout the text ask
for proofs of propositions. Reading this section when studying Sections 1.6-1.8, and then
periodically thereafter, will be rewarded, as your proof-writing ability grows.
• REFERENCES AND ADVICE ON THE WRITING PROJECTS
Near the end of this book you will find some general advice on the Writing Projects
given at the end of each chapter. There is a discussion of various resources available in the
mathematical sciences (such as Mathematical Reviews and the World Wide Web), tips on
doing library research, and some suggestions on how to write well. In addition, there
is a rather extensive bibliography of books and articles that will be useful when
researching these projects. We also provide specific hints and suggestions for each project,
with pointers to the references; these can be found in the solutions section of this manual,
at the end of each chapter.
• SAMPLE CHAPTER TESTS
Near the end of this book you will find a sequence of 13 chapter tests, comparable
to what might be given in a course. You can take these sample tests in a simulated test
setting as practice for the real thing. Complete solutions are provided, of course.
• PROBLEM-SOLVING TIPS AND LIST OF COMMON MISTAKES
People beginning any endeavor tend to make the same kinds of mistakes. This is
especially true in the study of mathematics. I have included a detailed list of common
misconceptions that students of discrete mathematics often have and the kinds of errors
they tend to make. Specific examples are given. It will be useful for you to review this list
from time to time, to make sure that you are not falling into these common traps. Also
included in this section is general advice on solving problems in mathematics, which you
will find helpful as you tackle the exercises.
• CRIB SHEETS
Finally, I have prepared a set of 13 single-page "crib sheets," one for each chapter
of the book. They provide a quick summary of all the important concepts, definitions, and
lll
