MC0063(B)

ModernApplicationsusingDiscreteMathematicalStructures

Contents

Unit1

Preliminaries 1

Unit2

TheoryofNumbersandIntroductiontoCryptography 23

Unit3

FormalLanguages 51

Unit4

BooleanAlgebrasandLogicalCircuits 69

Unit5

FiniteStateAutomata 104

Unit6

AlgebraicCodes 128

Unit7

FuzzySetsandFuzzyLogic 151

Unit8

Graphs 177

Unit9

TreesandAlgorithms 192

Edition:Fall2007

BKIDB067724thNovember2007

Unit10

Traversability 225

Unit11

Planarity,ColoringandPartitioning 234

Unit12

RepresentationsofGraphs 267

Unit13

DirectedGraphs 286

ModernApplicationsusingDiscreteMathematicalStructureslinkscomputer

science and mathematics. The foundation is pure mathematics and has

huge applications in different areas of Science and Technology. It is a

powerfultooltostudyandunderstandmanyconceptsofcomputerscience.

Astheadvanceoftechnologyisreliantonthegrowthofscience,andintern

science absolutely depends on mathematics, the study of Discrete

Mathematics through more significant. This book provides the reader a

comprehensiveideaaboutDiscreteMathematics.Themainaimofthebook

istopresent thefoundationsofcomputerrelatedconceptssothestudents

can understand specific computer science applications. Discrete

Mathematical structureshavemanyobjectives, few amongwhichare that

studentslearntheessentialsofmathematicstoacquirethelogicalthinking.

To accomplish theseobjectives, the version of this book easily explicable

andprovidesproblemsolvingtechniquesthroughselflearning,andalsothe

authorsmadeanextensiveuseofworkedexamplestodevelopthegeneral

ideas.Thestyleofpresentationofthelanguageissimpleandprecise.Most

of the symbols, notations used are standard. Suggestions for future

improvementsofthisbookwillbegratefullyacknowledged.

This book provided into thirteen units. The essence of each unit is given

below.

Unit1 dealswithsets,functions,equivalenceRelations,algebraicSystems

Algorithms.

Unit2 isdevoted tofundamentalconceptsofnumber theory,congruence,

arithmetical functions, method of repeated squares, some applications to

cryptography.

Unit3 covers formalLanguages, languagegeneratedby a grammarBNF

form,typesofGrammars.

SUBJECTINTRODUCTION

Unit 4 deals with Boolean Algebras, properties of Boolean Algebras,

Boolean Expressions, conjunctive and disjunctive normal forms, logical

switchingcircuitsandlogicgates.

Unit5explainsfinitestatemachines,finitestateautomata,statediagrams,

statetables,DFA,NDFA,Turingmachine.

Unit6coversalgebraiccodes,hammingdistance,linearcodes,paritycheck

codes,generatormatrices.

Unit7dealswithfuzzysets,fuzzyrelations,classicallogicandfuzzylogic,

linguisticvariable,fuzzytruthqualifier.

Unit8devotedtoGraphs,fundamentaldefinitions,Illustrations.

Unit 9 deals with trees, characterization of trees, rooted trees and

applications,spanningtrees,algorithmsforspanningtrees.

Unit 10 covers traversibility, Euler and Hamiltonian graphs, and traveling

salesmanproblem.

Unit11coversplanarrepresentationofagraph,dualgraphs,coloring,and

partitioning,findingachromaticnumber.

Unit12dealswithmatrixarrayrepresentationofgraphs,adjacencymatrix

andincidencematrix,circuitmatrix,pathmatrixandtheirproperties.

Unit 13 is devoted to directed graphs, definitions and examples, binary

relationasadigraphEulersdigraphs,matrixrepresentationofdigraphs.

Each unit is given in a detailed approach with appropriate illustrations.

Completeproofs,verificationsareprovidedwheneverneeded. Asufficient

numberofselfassessmentquestionsandanswers/hintsareprovided.

Brig.(Dr).R.S.GrewalVSM(Retd.)ProViceChancellorSikkimManipalUniversityofHealth,Medical&TechnologicalSciences

BoardofStudies1. Mr.RajuBPG 4. Dr.KaruppuSamy

Convener GMEmbeddedIntelligenceManipalUniversalLearning JupiterStrategicTechnologiesLtd.Manipal Bangalore

2. Mr.SunilKumarPandey 5. Mr.HarishchandraHebbarAsst.Professor DepartmentofITandCA DirectorMCISSikkimManipalUniversityDDE ManipalManipal

3. Dr.N.V.SubbaReddy 6. Mr.ArunC.MudholProfessor&HOD DepartmentofCS&E SolutionsOrientedProfessionalMIT,Manipal ITConsultant

Bangalore

ContentDevelopment1. Dr.KunchamSyamPrasad 2. Mr.DeepakShetty

AssociateProfessor AssistantProfessorDepartmentofMathematics SMU,ManipalMIT,Manipal

ConceptDesign&Editing1. Dr.BhavanariSatyanarayana 2. KedukodiBabushriSrinivas

Professor SeniorLecturerDepartmentofMathematics DepartmentofMathematicsAcharyaNagarjunaUniversity ManipalInstituteofTech.NagarjunaNagar522510 ManipalUniversityAndhraPradesh. Manipal

June,2007 ManipalUniversalLearningPvt.Ltd.,Manipal576104This book is a distance education module comprising a collection of learningmaterialforourstudents.All rights reserved. No part of this work may be reproduced in any form,bymimeographyoranyothermeans,withoutpermission inwriting fromSikkimManipalUniversity,Gangtok,Sikkim.PrintedandPublishedonbehalfofSikkimManipalUniversity,Gangtok,SikkimbyMr.RajkumarMascreen,GM,ManipalUniversalLearningPvt.Ltd.PrintedatManipalPressLimited,Manipal.

NotationsandSymbols

S.No. Symbol Description

1 a A

f or F

aisinthesetA

Emptyset

2 Thesetofnaturalnumbers

3. Thesetofintegers

4. Thesetofrationalnumbers

5. Thesetofrealnumbers

6. Setofcomplexnumbers

7. A B

A B

AisasubsetofB

AisapropersubsetofB

8. [a] Equivalenceclasscontaininga.

9 A B UnionofthesetsAandB

10. A B IntersectionofAandB

11. A1 ThecomplementofA

12 A B CartesianproductofAandB

13. f1 Inverseofthefunctionf

14. (X) PowersetofX

15. n(A) NumberofelementsinA

16. x A xisnotanelementofA

17. n Integersmodulon

18. Meet

19. join

20. +orZ+ Thesetofpositiveintegers

21. a b(modn) aiscongruenttobmodulon

22. Ifandonlyif

23. Productclass

24.

Floorfunction

Ceilingfunction

25. deg.v Degreeofthevertexv.