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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 self-study 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 proofs--things they do not usually see in calculus at present. This work by Oberste-Vorth (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 end-of-chapter 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
-
- Book Details
- Table of Contents
- Reviews
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 self-study 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 proofs--things they do not usually see in calculus at present. This work by Oberste-Vorth (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 end-of-chapter 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
