Table Of ContentG . P. G a v r il o v , A . A S ap o z h e nk o
S e l e c t e d P r o b l e m s i n
D I S C R E T E
M AT H E M AT I C S
M I R P U B L I S H E R S M O S C OW
selected
Problems In
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G.P. Gavrilov,
A.A. sapozhenko
Mir Publishers Moscow
Translated from Rus.siaJl by
Ram S Wadhwa and
NatalJa V Wadhv.a
FJrse pub1l5bed J 989
Rev 1.sed froru the t 977 u.ssian e
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trBllslat1on~ Puhl~sbers,
Contents
7
Preface
Chapter 1. Boolean Functions: Methods of Defining and
Basic Properties 10
Ll. Boolean Vectors and a Unit n-Dimensional Cube 10
i 2. Methods of Defining Boolean Functions. Elementary
Functions. Formulas. Superposition Operation 22
i 3 Special Forms of Formulas. Disjunctive and Con
junctive Normal Forms. Polynomials 33
i.4. Minimization of Boolean Functions 42
1.5 Essential and Apparent Variables 49
Chapter 2. Closed Classes and Completeness 55
2. i. Closure Operation. Closed Classes 55
2.2. Duality and the Class of Self-Dual Functions 59
2.3. Linearity and the Class of Linear Functions 63
2.4. Classes of Functions Preserving the Constants 67
2.5. 1\lonotonicity and the Class of Monotonic Functions 70
2.6. Completeness and Closed Classes 76
Chapter 3. k· Valued Logics 82
3.1. Representation of Functions of k-Valued Logics
Through Formulas 82
3.2. Closed Classes and Completeness in k-Valued
Logics 88
Chapter 4. Graphs and Networks
101
4.1. Basic Concepts in the Graph Theory 101
4.2. Planarity, Conn~>ctivity, and Numerical Charac-
teristics Qt Graphs
H{}
4.3. Directed Graphs
117
4.4. Trees and Bipolar Networks
123
4.5. Estimates in the Theory of Graphs and Networks
1.37
4.6. Representations of Boolean Functions by Contact
Schemes and Formulas
143
CONTE'lTS
Chapter 5 Fundamentals of Codlng Theoey t5S
5 \ Codes w1tb 1.55
Co-rretllon~
5 2 Linear Codes 160
a
5 A1phabe\tc Cod1ng 1'03
Chapter 6 Fi.o I te .Automatons
174
6 t Determinate and Boundedly D:eterm1nate Functions 174
6 2 Representation ol Determinate Funct1ons b_y Afoore
D1agrams Canontcal Equat1ons Tatdes.and Schemes
Operations In volvtng Detetm1na te Funetlons i87
6 3 Closed and Completeness the Sets of
Cla~ses In
Determanatt! and BlJundedly Determtnale Fune
206
llODS
Chapter 7 Fundamr;ntRls of the Algorithm Theory 212
1 I Tur1Dg s },faehln~s and Opera. t1ons "all th Them
Funt1lons Cnmputahl~ an Turlng ! l\.1 ath1n~s 212
1 2 ol Computable and Recurs1'e Functions 233
Classe~
7 3 ComputabJltty and of ComputatJons
Comple1:~ty
2~1
Chapler 8 Elements ol Combioat<-riaJ Analy5m 248
8 J Permutations and CombtnatJons Propert1es of Bu1o
ml al Coeflic len ts 2.48
8 2 I ncl ws ion and E .ze) us Ion Formulas
259
8 8 Recurrent Sequencee Generat1og Funetlans and
Recurrence ReJal1ons
265
8 4 Pol)A s Theory Z75
8 5 Asymptot 1c E 1press1ons .~&nd Inequallttes .280
Sol u.tions, Answers and Hints
289
t
B 1bJ 10gra phy
403
Notations
405
S11bjec l Index
40!1
Preface
This collection of problems is intended as an accompani
ment to a course on discrete mathematics at the universi
ties. Senior students and graduates specializing in mathe
matical cybernetics may also find the book useful. Lectur
ers can use the material for exercises during seminars.
The material in this book is based on a course of lec
tures on discrete mathematics delivered by the authors over
-
a number of vears at the Facultv. of Mechanics and Mathe-
matics, and later at the Faculty of Computational Mathe-
matics and Cybernetics at i\loscow State University.
The reader can use Introduction to Discrete
~1fathema
tics by S. Yablonsky as the main text. when solving the
problems in this collection.
The book consists of eight. chaptet·s. The ftrst two chap
ters are devoted to Boolean algebra which forms the ba
sis of discrete mathematics. About a quarter of the total
teaching time during lectures and practica1s at the Com
putationall\fathelllatics and Cybernetics Facult.y at Mos
cow University is devoted to Boolean algebra. The ma
terial in this part introduces the student to the concepts
of discrete. functions, superposition, and functionally
complete sets. It also acquaints the student with various
methods for specifying a discrete function (tables, poly
nomial representation, normal forms, geometrical repre
sentation using an n-dimensional unit cube, etc.). Me
thods for testing the completeness and closure of sets of
functions are also considered.
The third chapter is devoted to k-valued logics. The
problems presented are intended to acquaint the reader
with the canonical expansions ot k-valued functions, equiv
alent transformations of formulas, closed classes of the
k-valued functions, and methods for testing the complete
ness and closure of functions. Several problems in the
PREFACE
8
ehapter 1llustrate the d1ilerence betv.een k valued log1cs
>
(k 2) and Boolean algebra
The fourth chapter conta1ns problems on the theory of
dlrected and undtrected graphs and the network and ctrcutt
theory The chapter descr1bes the baste concepts methods
and terms of graph theory v..h1ch are 1.1sed to de
w1d~ly
scr1he and 111Vest1gate the structural propert1es of obJects
In various branches of sctence and technology TJ1e prob
le-ms a.re intended t.o eonsolldate the bas1c concepts of
graph theory to Illustrate the application of network and
graph theory to the construction of cJrcuJts epresent1ng
1
a
Boolean functtons to count the number of obJects w1th
gtven geometrJcal structure etc The authors hope that
the lecturer wtll also find problems thts chapter to
1n
help h1m demonstrate the mathemattcal r1gor dur1ng the
proof of geometrtcally ob-v1ous statements
'rhe fi(tb chapter the baste concepts o[ c.od1ng
des~tlbeS
tlteory The problems concern the properties of error cor
rect1ng codes, alphabetJcal codes and m1n1mum redun
dancy codes
The chapter conta1ns problems demonstrating
SI>..tii
d1fierent ways of descr1btng d1screte transformers (auto
matons) Problems a1med at revealtng deternltntst•c and
boundedly determinl.SlJc automatons are also gtven Other
problems the d1fierent ways of representing auto
conc.~l"n
matons (diagrams canonical equations and schemes (ctr
CUlts)) the 1nvestJgat1on ol the funct1onal completeness
and closure of sets of autontaton mapptngs and also the
properties of operations Involving such mappJngs
The seventh chapter deals the elements oi algo
Witll
rithm theory and 1s Intended to provide an 1dea about
effect1ve computabtl1ty and complexity of computations
lt 18 also about eertaln ways fot spec1fy1ng algortthms
suclt as Tur1ng s maeh1oes and recursive functions
!he e1ghtb chapter descr1bes the elements of co.mbtna
tor1al \Vhtle studytng dtscrete mathemattcs
~nalys1s
one frequently comes across questtons concerning the
ex1stence eount1ng and est•mat1on of v arJous combtna
tor1al obJects Hence comblnator,al problems are
10
eluded book
In the
For the sake a[ c.onven1ence authors have started
th~
each 5ecl1on uttb a theoretical background
H1nts and answers are prov1ded far most (but not c;J.ll)
PREFACE 9
problems. Solutions are given in a concise form in the
form of notes, and trivial conclusions are omitted. In
some cases, only the outlines of solutions are presented.
The exercises in the book have various origins. Most of
the material is traditional and specialists on discrete
mathematics are all too familiar with such problems.
However, it is practically impossible to trace the origin
of the problems of this kind. l\Iost of the problems were
conceived by the authors during seminars and practical
classes, during examinations, and also while preparing
this hook. Some of the problems resulted from studying
publications in journals, and a few have been borrowed
from other sources. Sevel·al problems were passed on to
us by staff at the Faculty and by other colleagues. The
authors express their sincere gratitude to them all.
The authors are deeply indebted to S. V. Yablonsky
for his persistent interest during the preparation of this
book. His comments and suggestions played a significant
role in determining the structure and scope of this book.
We are also grateful to our reviewers V .V. Glagolev
and A.A. Markov for their critica.l comments and sugges
tions for improving the collection.
G.P. Gavrilov
A.A. Sapozhenko
