
Book DetailsAMS/MAA TextbooksVolume: 22; 2012; 232 pp
Reprinted edition available: TEXT/56
A Bridge to Abstract Mathematics will prepare the mathematical novice to explore the universe of abstract mathematics. Mathematics is a science that concerns theorems that must be proved within the constraints of a logical system of axioms and definitions rather than theories that must be tested, revised, and retested. Readers will learn how to read mathematics beyond popular computational calculus courses. Moreover, readers will learn how to construct their own proofs. The book is intended as the primary text for an introductory course in proving theorems, as well as for selfstudy or as a reference. Throughout the text, some pieces (usually proofs) are left as exercises. Part V gives hints to help students find good approaches to the exercises. Part I introduces the language of mathematics and the methods of proof. The mathematical content of Parts II through IV were chosen so as not to seriously overlap the standard mathematics major. In Part II, students study sets, functions, equivalence and order relations, and cardinality. Part III concerns algebra. The goal is to prove that the real numbers form the unique, up to isomorphism, ordered field with the least upper bound. In the process, we construct the real numbers starting with the natural numbers. Students will be prepared for an abstract linear algebra or modern algebra course. Part IV studies analysis. Continuity and differentiation are considered in the context of time scales (nonempty, closed subsets of the real numbers). Students will be prepared for advanced calculus and general topology courses. There is a lot of room for instructors to skip and choose topics from among those that are presented.

Table of Contents

front cover

copyright page

title page

Contents

Some Notes on Notation

To the Students

To Those Beginning the Journey into Proof Writing

How to Use This Text

Do the Exercises!

Acknowledgments

For the Professors

To Those Leading the Development of Proof Writing for Students in a Broad Range of Disciplines

I THE AXIOMATIC METHOD

Introduction

The History of Numbers

The Algebra of Numbers

The Axiomatic Method

Parallel Mathematical Universes

Statements in Mathematics

Mathematical Statements

Mathematical Connectives

Symbolic Logic

Compound Statements in English

Predicates and Quantifiers

Supplemental Exercises

Proofs in Mathematics

What is Mathematics?

Direct Proof

Contraposition and Proof by Contradiction

Proof by Induction

Proof by Complete Induction

Examples and Counterexamples

Supplemental Exercises

How to THINK about mathematics: A Summary

How to COMMUNICATE mathematics: A Summary

How to DO mathematics: A Summary

II SET THEORY

Basic Set Operations

Introduction

Subsets

Intersections and Unions

Intersections and Unions of Arbitrary Collections

Differences and Complements

Power Sets

Russell's Paradox

Supplemental Exercises

Functions

Functions as Rules

Cartesian Products, Relations, and Functions

Injective, Surjective, and Bijective Functions

Compositions of Functions

Inverse Functions and Inverse Images of Functions

Another Approach to Compositions

Supplemental Exercises

Relations on a Set

Properties of Relations

Order Relations

Equivalence Relations

Supplemental Exercises

Cardinality

Cardinality of Sets: Introduction

Finite Sets

Infinite Sets

Countable Sets

Uncountable Sets

Supplemental Exercises

III NUMBER SYSTEMS

Algebra of Number Systems

Introduction: A Road Map

Primary Properties of Number Systems

Secondary Properties

Isomorphisms and Embeddings

Archimedean Ordered Fields

Supplemental Exercises

The Natural Numbers

Introduction

Zero, the Natural Numbers, and Addition

Multiplication

Supplemental Exercises

Summary of the Properties of the Nonnegative Integers

The Integers

Introduction: Integers as Equivalence Classes

A Total Ordering of the Integers

Addition of Integers

Multiplication of Integers

Embedding the Natural Numbers in the Integers

Supplemental Exercises

Summary of the Properties of the Integers

The Rational Numbers

Introduction: Rationals as Equivalence Classes

A Total Ordering of the Rationals

Addition of Rationals

Multiplication of Rationals

An Ordered Field Containing the Integers

Supplemental Exercises

Summary of the Properties of the Rationals

The Real Numbers

Dedekind Cuts

Order and Addition of Real Numbers

Multiplication of Real Numbers

Embedding the Rationals in the Reals

Uniqueness of the Set of Real Numbers

Supplemental Exercises

Cantor's Reals

Convergence of Sequences of Rational Numbers

Cauchy Sequences of Rational Numbers

Cantor's Set of Real Numbers

The Isomorphism from Cantor's to Dedekind's Reals

Supplemental Exercises

The Complex Numbers

Introduction

Algebra of Complex Numbers

Order on the Complex Field

Embedding the Reals in the Complex Numbers

Supplemental Exercises

IV TIME SCALES

Time Scales

Introduction

Preliminary Results

The Time Scale and its Jump Operators

Limits and Continuity

Supplemental Exercises

The Delta Derivative

Delta Differentiation

Higher Order Delta Differentiation

Properties of the Delta Derivative

Supplemental Exercises

V HINTS

Hints for (and Comments on) the Exercises

Hints for Chapter 2

Hints for Chapter 3

Hints for Chapter 4

Hints for Chapter 5

Hints for Chapter 6

Hints for Chapter 7

Hints for Chapter 8

Hints for Chapter 9

Hints for Chapter 10

Hints for Chapter 11

Hints for Chapter 12

Hints for Chapter 13

Hints for Chapter 14

Hints for Chapter 15

Hints for Chapter 16

Bibliography

Index

About the Authors


Reviews

For a variety of reasons, over the past 30 years or so, "bridge" or "transition" courses have become staples in the undergraduate mathematics curriculum. The purpose of these courses, broadly speaking, is to introduce students to abstract and rigorous mathematical thinking, at a level appropriate to their learning, to make conjectures and construct proofsthings they do not usually see in calculus at present. This work by ObersteVorth (Indiana State), Mouzakitis (Second Junior High School of Corfu, Greece), and Lawrence (Marshall Univ.) has evolved from courses taught at the University of South Florida and Marshall University and is worthy of consideration. Coverage includes standard ideas involving set, functions, relations, and cardinality as well as mathematical statements and logic and types of proof. Building on these early notions, an instructor can then choose to go in the direction of number systems (including construction of the reals from the rationals) with an algebraic flavor or toward analysis (here, including time scales and continuity). The analysis direction is perhaps the rockier road to travel. Given the purpose and the audience, the exposition is commendably open and not terse. The book includes scores of exercises scattered throughout, with many endofchapter supplemental exercises.
D. Robbins, CHOICE 
To begin the process of being able to write and understand proofs it is necessary for the student to go back a few squares on the mathematical board game and learn the rigorous definitions of concepts such as the structure of mathematical statements, set theory and the underlying structural definitions of the basic number systems. Knowing these concepts very well gives the student the foundation for entering the proof realm and it helps to overturn their complacent belief of understanding. This book is designed to give the reader that understanding and the mission is a success. The authors provide detailed explanations of the foundations of mathematics needed to work comfortably with proofs, both operationally and theoretically. It would be an excellent choice for a freshman/sophomore level course in the foundations of mathematics designed to prepare students for the rigors of proofs that they will experience in their later years.
Charles Ashbacher, Journal of Recreational Mathematics


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Reprinted edition available: TEXT/56
A Bridge to Abstract Mathematics will prepare the mathematical novice to explore the universe of abstract mathematics. Mathematics is a science that concerns theorems that must be proved within the constraints of a logical system of axioms and definitions rather than theories that must be tested, revised, and retested. Readers will learn how to read mathematics beyond popular computational calculus courses. Moreover, readers will learn how to construct their own proofs. The book is intended as the primary text for an introductory course in proving theorems, as well as for selfstudy or as a reference. Throughout the text, some pieces (usually proofs) are left as exercises. Part V gives hints to help students find good approaches to the exercises. Part I introduces the language of mathematics and the methods of proof. The mathematical content of Parts II through IV were chosen so as not to seriously overlap the standard mathematics major. In Part II, students study sets, functions, equivalence and order relations, and cardinality. Part III concerns algebra. The goal is to prove that the real numbers form the unique, up to isomorphism, ordered field with the least upper bound. In the process, we construct the real numbers starting with the natural numbers. Students will be prepared for an abstract linear algebra or modern algebra course. Part IV studies analysis. Continuity and differentiation are considered in the context of time scales (nonempty, closed subsets of the real numbers). Students will be prepared for advanced calculus and general topology courses. There is a lot of room for instructors to skip and choose topics from among those that are presented.

front cover

copyright page

title page

Contents

Some Notes on Notation

To the Students

To Those Beginning the Journey into Proof Writing

How to Use This Text

Do the Exercises!

Acknowledgments

For the Professors

To Those Leading the Development of Proof Writing for Students in a Broad Range of Disciplines

I THE AXIOMATIC METHOD

Introduction

The History of Numbers

The Algebra of Numbers

The Axiomatic Method

Parallel Mathematical Universes

Statements in Mathematics

Mathematical Statements

Mathematical Connectives

Symbolic Logic

Compound Statements in English

Predicates and Quantifiers

Supplemental Exercises

Proofs in Mathematics

What is Mathematics?

Direct Proof

Contraposition and Proof by Contradiction

Proof by Induction

Proof by Complete Induction

Examples and Counterexamples

Supplemental Exercises

How to THINK about mathematics: A Summary

How to COMMUNICATE mathematics: A Summary

How to DO mathematics: A Summary

II SET THEORY

Basic Set Operations

Introduction

Subsets

Intersections and Unions

Intersections and Unions of Arbitrary Collections

Differences and Complements

Power Sets

Russell's Paradox

Supplemental Exercises

Functions

Functions as Rules

Cartesian Products, Relations, and Functions

Injective, Surjective, and Bijective Functions

Compositions of Functions

Inverse Functions and Inverse Images of Functions

Another Approach to Compositions

Supplemental Exercises

Relations on a Set

Properties of Relations

Order Relations

Equivalence Relations

Supplemental Exercises

Cardinality

Cardinality of Sets: Introduction

Finite Sets

Infinite Sets

Countable Sets

Uncountable Sets

Supplemental Exercises

III NUMBER SYSTEMS

Algebra of Number Systems

Introduction: A Road Map

Primary Properties of Number Systems

Secondary Properties

Isomorphisms and Embeddings

Archimedean Ordered Fields

Supplemental Exercises

The Natural Numbers

Introduction

Zero, the Natural Numbers, and Addition

Multiplication

Supplemental Exercises

Summary of the Properties of the Nonnegative Integers

The Integers

Introduction: Integers as Equivalence Classes

A Total Ordering of the Integers

Addition of Integers

Multiplication of Integers

Embedding the Natural Numbers in the Integers

Supplemental Exercises

Summary of the Properties of the Integers

The Rational Numbers

Introduction: Rationals as Equivalence Classes

A Total Ordering of the Rationals

Addition of Rationals

Multiplication of Rationals

An Ordered Field Containing the Integers

Supplemental Exercises

Summary of the Properties of the Rationals

The Real Numbers

Dedekind Cuts

Order and Addition of Real Numbers

Multiplication of Real Numbers

Embedding the Rationals in the Reals

Uniqueness of the Set of Real Numbers

Supplemental Exercises

Cantor's Reals

Convergence of Sequences of Rational Numbers

Cauchy Sequences of Rational Numbers

Cantor's Set of Real Numbers

The Isomorphism from Cantor's to Dedekind's Reals

Supplemental Exercises

The Complex Numbers

Introduction

Algebra of Complex Numbers

Order on the Complex Field

Embedding the Reals in the Complex Numbers

Supplemental Exercises

IV TIME SCALES

Time Scales

Introduction

Preliminary Results

The Time Scale and its Jump Operators

Limits and Continuity

Supplemental Exercises

The Delta Derivative

Delta Differentiation

Higher Order Delta Differentiation

Properties of the Delta Derivative

Supplemental Exercises

V HINTS

Hints for (and Comments on) the Exercises

Hints for Chapter 2

Hints for Chapter 3

Hints for Chapter 4

Hints for Chapter 5

Hints for Chapter 6

Hints for Chapter 7

Hints for Chapter 8

Hints for Chapter 9

Hints for Chapter 10

Hints for Chapter 11

Hints for Chapter 12

Hints for Chapter 13

Hints for Chapter 14

Hints for Chapter 15

Hints for Chapter 16

Bibliography

Index

About the Authors

For a variety of reasons, over the past 30 years or so, "bridge" or "transition" courses have become staples in the undergraduate mathematics curriculum. The purpose of these courses, broadly speaking, is to introduce students to abstract and rigorous mathematical thinking, at a level appropriate to their learning, to make conjectures and construct proofsthings they do not usually see in calculus at present. This work by ObersteVorth (Indiana State), Mouzakitis (Second Junior High School of Corfu, Greece), and Lawrence (Marshall Univ.) has evolved from courses taught at the University of South Florida and Marshall University and is worthy of consideration. Coverage includes standard ideas involving set, functions, relations, and cardinality as well as mathematical statements and logic and types of proof. Building on these early notions, an instructor can then choose to go in the direction of number systems (including construction of the reals from the rationals) with an algebraic flavor or toward analysis (here, including time scales and continuity). The analysis direction is perhaps the rockier road to travel. Given the purpose and the audience, the exposition is commendably open and not terse. The book includes scores of exercises scattered throughout, with many endofchapter supplemental exercises.
D. Robbins, CHOICE 
To begin the process of being able to write and understand proofs it is necessary for the student to go back a few squares on the mathematical board game and learn the rigorous definitions of concepts such as the structure of mathematical statements, set theory and the underlying structural definitions of the basic number systems. Knowing these concepts very well gives the student the foundation for entering the proof realm and it helps to overturn their complacent belief of understanding. This book is designed to give the reader that understanding and the mission is a success. The authors provide detailed explanations of the foundations of mathematics needed to work comfortably with proofs, both operationally and theoretically. It would be an excellent choice for a freshman/sophomore level course in the foundations of mathematics designed to prepare students for the rigors of proofs that they will experience in their later years.
Charles Ashbacher, Journal of Recreational Mathematics