MTH202 Lecture Notes

Preface

This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, Business, and the sciences. Besides reading the book, students are strongly encouraged to do all the excercises. Mathematics is a discipline in which working the problems is essential to the understanding of the material contained in this book. Students are encouraged first to do the problems without referring to the solutions. Solutions to problems found at the end of this book can only be used when you are stuck. Exert a reasonable amount of efforts towards solving a problem before you look up the answer, and rework any problem you miss. Students are strongly encouraged to keep up with the exercises and the sequel of concepts as they are going along, for mathematics builds on itself. Solution Manual for the text can be requested from the author through

May 2001

Contents

Preface

2 Conditional and Biconditional Propositions 3 Rules of Inferential Logic . . . . . . . . . . . . . . . . . . . . . . . 4 Propositions and Quantifiers . . . . . . . . . . . . . . . . . . . . . 5 Arguments with Quantified Premises . . . . . . . . . . . . . . . . 6 Project I: Digital Logic Design 7 Project II: Number Systems . . . . . . . . . . . . . . . . . . . . .

Fundamentals of Mathematical Proofs

8 Methods of Direct Proof I . . . . . . . . . . . . . . . . . . . . . . 9 More Methods of Proof 10 Methods of Indirect Proofs: Contradiction and Contraposition 11 Method of Proof by Induction . . . . . . . . . . . . . . . . . . . 12 Project III: Elementary Number Theory and Mathematical Proofs 74 13 Project IV: The Euclidean Algorithm . . . . . . . . . . . . . . . 14 Project V: Induction and the Algebra of Matrices

Fundamentals of Set Theory

15 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Properties of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Project VI: Boolean Algebra . . . . . . . . . . . . . . . . . . . .

Relations and Functions

18 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 19 Partial Order Relations . . . . . . . . . . . . . . . . . . . . . . . 110 CONTENTS

20 Functions: Definitions and Examples . . . . . . . . . . . . . . . 116 21 Bijective and Inverse Functions . . . . . . . . . . . . . . . . . . . 124 22 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 23 Project VII: Applications to Relations . . . . . . . . . . . . . . . 145 24 Project VIII: Well-Ordered Sets and Lattices . . . . . . . . . . . 148 25 Project IX: The Pigeonhole Principle . . . . . . . . . . . . . . . 149 26 Project X: Countable Sets . . . . . . . . . . . . . . . . . . . . . 150 27 Project XI: Finite-State Automaton . . . . . . . . . . . . . . . . 152

Introduction to the Analysis of Algorithms

28 Time Complexity and O-Notation . . . . . . . . . . . . . . . . . 157 29 Logarithmic and Exponential Complexities . . . . . . . . . . . . 165 30 Θand Ω-Notations . . . . . . . . . . . . . . . . . . . . . . . . . 169

Fundamentals of Counting and Probability Theory

31 Elements of Counting . . . . . . . . . . . . . . . . . . . . . . . . 173 32 Basic Probability Terms and Rules . . . . . . . . . . . . . . . . . 180 33 Binomial Random Variables . . . . . . . . . . . . . . . . . . . . 191

Elements of Graph Theory

34 Graphs, Paths, and Circuits . . . . . . . . . . . . . . . . . . . . 197 35 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211