Fundamentals of Elementary Mathematics

Preface

Acknowledgments

Glossary of Symbols


1 An Introduction to Logic and Mathematical Reasoning


1.1 Basic Ideas


1.2 Forming New Statements from Given Statements


1.3 Negation, Disjunction, and Conjunction


1.4 Conditional Statements


1.5 The Biconditional, Logical Equivalence, Tautologies, and Contradictions


1.6 Statements Related to a Conditional


1.7 Some Important Properties


1.8 Open Sentences and Quantifiers


1.9 The Negation of Universally and Existentially Quantified Statements


1.10 Some Rules of Inference—The Tools of Proof


1.11 Summary of Chapter 1


2 Sets, Relations, Functions, and Operations


2.1 The Development of Mathematics


2.2 More on the Set Concept and Set Notation


2.3 Relations on Sets


2.4 Venn Diagrams


2.5 One-to-One Correspondence


2.6 Operations on Sets


2.7 Element Tables


2.8 Algebra of Sets


2.9 Relations


2.10 Reflexive, Symmetric, Transitive, and Equivalence Relations


2.11 Functions


2.12 Operations


2.13 Mathematical Systems and Isomorphism


2.14 Summary of Chapter 2


3 The System of Whole Numbers


3.1 Introduction


3.2 Construction of the Nonempty Set—The Set of Whole Numbers


3.3 Equality and Addition in W


3.4 Structural Properties of +


3.5 Multiplication in W and Structural Properties of ×


3.6 Order in the Whole Numbers


3.7 Subtraction and Division in W


3.8 Some Comments on Multiplication and Division in W


3.9 Counting, Cardinal and Ordinal Use of Whole Numbers


3.10 Elementary Number Theory


3.11 The Whole-Number Line, Open Sentences, Graphs


3.12 The Whole-Number Plane, Relations on W, Open Sentences in Two Variables


3.13 Summary of Chapter 3


4 Numeration System


4.1 Introduction


4.2 The Additive and Positional Principles


4.3 Examples of Numeration Systems


4.4 Positional Systems and the Hindu-Arabic System


4.5 Positional Numeration Systems with Bases Other than Ten


4.6 The Babylonian Numeration System—A Sexagesimal System


4.7 Arithmetic in Other Bases


4.8 Positional Numerals for Fractional Numbers


4.9 Summary of Chapter 4


5 Algorithms For Computation with Whole Numbers


5.1 Introduction


5.2 Addition Algorithms


5.3 Subtraction Algorithms


5.4 Multiplication Algorithms


5.5 Division Algorithms

Summary of Chapter 5


6 The System of Fractional Numbers


6.1 Introduction


6.2 Construction of the Fractional-Number System


6.3 Addition and Multiplication in F


6.4 Structural Properties of ⊞ and ⊡


6.5 Fractions and Standard Names for Fractional Numbers


6.6 An Isomorphism Between the System of Whole Numbers and a Sub System of the Fractional Number


6.7 Mixed Numerals


6.8 Order, Subtraction, and Division in F


6.9 Density in F, a One-To-One Correspondence Between W and F


6.10 Interpretations of Fractional Number


6.11 The Fractional-Number Line and Plane, Open Fractional-Number Sentences, Graphs of Subsets of F and F × F


6.12 Summary of Chapter 6


7 The System of Integers


7.1 Introduction


7.2 Construction of the System of Integers


7.3 Addition and Multiplication in I


7.4 Positive and Negative Integers, Standard Names


7.5 Structural Properties of ⊕ and ⊗


7.6 An Isomorphism Between W and the Nonnegative Integers


7.7 Subtraction, Division, Order, Absolute Value in I


7.8 Elementary Number Theory


7.9 A Brief Section on Open Integer Sentences


7.10 Summary of Chapter 7


8 The System of Rational Numbers


8.1 Introduction


8.2 Construction of the Rational-Number System


8.3 Addition and Multiplication in R


8.4 Structural Properties of ⨹ and ◬


8.5 Positive and Negative Rational Numbers, Rational Numerals—The Standard Names for Rational Numbers


8.6 An Isomorphism Between the System of Integers and a Subsystem of the Rationals


8.7 Subtraction, Division, Order in R


8.8 An Interpretation of Rational Numbers


8.9 Open Number Sentences, Graphs


8.10 Summary of Chapter 8


9 Decimal Numbers Introduction: Real Numbers


9.1 Introduction


9.2 Decimal Numerals for Fractional Numbers and Computational Algorithms Using Decimal Numerals


9.3 One-to-One Correspondence Between Fractional Numbers and Repeating Decimals


9.4 Repeating Decimals Which Correspond to Rational Numbers


9.5 Infinite Decimals and the Real Numbers


9.6 A Brief Overview of the Number Systems


9.7 Mathematical Sentences


9.8 Summary of Chapter 9


10 Abstract Systems


10.1 Introduction


10.2 Groups


10.3 Functions and Groups


10.4 Definition of Ring and Field


10.5 Congruence Modulo B—Finite Rings and Fields


10.6 Open Linear Congruences


10.7 Summary of Chapter 10

Answers and Suggestions for Selected Exercises

Index