Table Of ContentTexts in Computer Science
Sergei Kurgalin
Sergei Borzunov
The Discrete
Math Workbook
A Companion Manual for Practical Study
Texts in Computer Science
Series editors
David Gries, Department of Computer Science, Cornell University, Ithaca, NY,
USA
Orit Hazzan, Faculty of Education in Science and Technology, Technion—Israel
Institute of Technology, Haifa, Israel
More information about this series at http://www.springer.com/series/3191
Sergei Kurgalin (cid:129) Sergei Borzunov
The Discrete Math
Workbook
A Companion Manual for Practical
Study
123
SergeiKurgalin SergeiBorzunov
Voronezh State University Voronezh State University
Voronezh, Russia Voronezh, Russia
ISSN 1868-0941 ISSN 1868-095X (electronic)
Textsin Computer Science
ISBN978-3-319-92644-5 ISBN978-3-319-92645-2 (eBook)
https://doi.org/10.1007/978-3-319-92645-2
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Preface
Rapid development of information and communication technologies is a charac-
teristic feature of the current time period. Training specialists who meet the
requirements of the time in the field of such technologies leads to the necessity of
payingspecialattentiontotheirmathematicaleducation.Herewithitisdesirableto
shapeknowledgepresentedintheformofpracticalpartofmathematicalcoursesin
such a way that after performing these tasks, students get the required competen-
cies.Oneofthemathematicalcourseswhichistaught,asarule,inearlysemesters
is the course of discrete mathematics. A number of general courses and specialist
disciplines included in the cycle of specialist training in the field of information
technologies are based on the present course. Thus, mathematical logic and theory
of algorithms form the theoretical foundation of informatics; Boolean algebra
servesasabasisformethodsofelectroniccircuitdevelopment;theoryofgraphsis
usedwhenconstructingmultiprocessorcomputingsystemsandcomputernetworks,
andinprogramming.Toteachstudents,thecompetentpracticalapplicationoftheir
theoreticalknowledgeisoneofthemethodologicalobjectivesthegivenstudyguide
can help with.
The study guide was created over a number of years and is backed up by the
teaching experience of the course at the Department of Computer Science of
Voronezh State University. Its contents correspond to Federal State Educational
Standards on the following training programs: “Information Systems and Tech-
nologies,” “Program Engineering,” and “Radiophysics.” The existing problem
books on discrete mathematics and mathematical logic do not fully cover the
necessary volume and information scope upon the above-mentioned programs to
graduate fully skilled subject matter experts; therefore, this study guide provides a
largenumberofproblemsofawiderangeofdifficultyforthesetrainingprograms.
The present study guide is intended for practical training, laboratory practicals,
and self-study. It contains basic theoretical concepts and methods of solving the
most fundamental problems; it forms the perceptions of sets, relations on sets, and
behavior of different types of relations, as well as functions, general concepts of
combinatorics,theoryofgraphs,Booleanalgebra,basicprinciplesofthealgorithms
theory,etc.Somechaptersofthestudybook,suchasChaps.10and11gobeyond
the traditional course of discrete mathematics. Nevertheless, we believe it is nec-
essary to include them into the structure of the study guide, as they promote
v
vi Preface
understandingofthemethodsofconstructionandanalysisofalgorithms.According
tothefamousChurch–Turingthesis,theTuringmachineisabletoimitateallmeans
of step-by-step computation and is considered to be the model of any computing
system existing today. The theory of algorithms’ asymptotic analysis studies the
methods of getting asymptotic estimations of algorithms’ computational complex-
ity,whichhastheparamountimportancefortheestimationoftheneedinresource
requirement at the specific implementation of algorithm.
It is only natural that we analyzed certain textbooks and problem books while
preparingthestudyguidewithaviewtothepositiveexperiencecontainedinthem
and to produce the most efficient method of presenting the material. Most of them
are enumerated in the reference list; in particular, these are [4, 15, 25].
Thestudyguidecoverstwosemestersofthestudycourse;somechaptersmight
be used in one-semester courses.
Eachchapterbeginswithatheoreticalpart,whichalbeitoccupieslittlespace,is
of crucial importance as it reviews the basic concepts of the course and sets the
utilizedterminology.Alongsidewiththebasicconcepts,theoreticalprovisions,and
formulae in the text, there are also instructions on their practical use. Then the
detailedsolutionofseveralmostcommonproblemsisshown,andtheproblemsfor
solvinginaclassroom(acomputerlaboratory)aredistributed.Theycanbeusedfor
self-study.Youwillfindtheanswersandsolutionstoalltheproblemsexceptforthe
simplest ones. Exercises differ in complexity level. The most difficult ones are
markedwithanasterisk(*),placednexttotheexercisenumber.Iftheoreticalpartis
provided with an example, its end is designated by the symbol h.
The differential characteristics of the study guide, in our opinion, are: detailed
problem solution containing typical methods and computational schemes; the
presence of a big amount of exercises and examples of different complexity level;
reflecting the subject field of information and communication technologies in
problemdefinitionswherewethoughtitrelevant.Moreover,takingintoaccountthe
specificity of bachelor’s training program in the field of such technologies, it is
neededtoimplementadevelopedalgorithminaparticularprogramminglanguage.
InChaps.9and12,fragmentsofcodeinthePascallanguagearesetinorderto
demonstrate the applicability of studied methods in programming practice. Head-
ingsofsomeproblemsinChap.12areinsemi-bold;theabove-mentionedproblems
are recommended to do in a computer laboratory.
Aseparatechapterisdedicatedtoparallelalgorithms.Multiprocessorcomputing
systems are extensively used at present for solving tasks on mathematical and
computer modeling,managementofdatabaseandcomplexsoftwarepackages,etc.
Furthermore, the majority of modern computing systems back up parallel compu-
tations on a hardware level. From this perspective, questions connected with con-
struction and analysis of parallel algorithms are becoming increasingly relevant. It
shouldbepointedoutthatinmanycases,developinganeffectiveparallelalgorithm
ofsolutionofsometaskrequiresattractionofnewideasandmethodscomparingto
creating a sequential algorithm version. These are, for instance, practically
importantproblemsofsearchingatargetelementindatastructures,evaluationofan
algebraic expression, etc. Theoretical part of this chapter takes a relatively bigger
Preface vii
place in comparison with other chapters, as it deals with more complex issues of
applied aspects of parallel programming, rather difficult when first studied. This
chapter requires good command of methods of problem solution described in the
previous chapters and has, in the authors’ opinion, increased complexity.
There is a sufficient amount of illustrations in the book, which help students
visualizeallobjectsunderconsiderationandconnectivitybetweenthem.Inthefinal
section, you will find reference data which include the Greek alphabet, basic
trigonometric formulae, brief summary on differential and integral calculi, and the
most important finite summas, which will mitigate the need for resorting to spe-
cialistreferenceliterature.Thiseditionisprovidedwiththereferencelist,aswellas
thenameandthesubjectindices.Afterthefirstmentioningofscientists’surnames,
consult the references with their short biographies taken from [82].
Below you can see the scheme of the chapter information dependence in the
form of an oriented graph reflecting the preferable order of covering the academic
material.Forinstance,afterhavingstudiedChaps.1,2,and3,youcanmovetoone
of the three chapters: 4, 5, or 6, in which contents are relatively independent. The
dashed border marks the chapters which will be suitable for readers who want to
reach a high level of the subject knowledge; the present sections contain more
difficultacademicmaterial.Thisway,afterstudyingChap.6youcaneithercometo
Chap. 9, or study the material of Chaps. 7 and 8 more thoroughly for better
digestion, and only then switch to Chap. 9.
Voronezh, Russia Sergei Kurgalin
Sergei Borzunov
viii Preface
Chapter 1
Chapter 2
Chapter 3
Chapter 4 Chapter 6 Chapter 5
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Thechapterdependencychart
Contents
1 Fundamentals of Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4 Combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.2 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.1 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.3 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.1 Karnaugh Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.2 Combinational Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.4 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.2 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
8.2 Answers, Hints, Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
ix
