**AMS/MAA Textbooks**

Volume: 22;
2012;
232 pp;
Hardcover

**Print ISBN: 978-0-88385-779-3
Product Code: TEXT/22**

List Price: $62.00

AMS Member Price: $46.50

MAA Member Price: $46.50

**Electronic ISBN: 978-1-61444-606-4
Product Code: TEXT/22.E**

List Price: $62.00

AMS Member Price: $46.50

MAA Member Price: $46.50

# Bridge to Abstract Mathematics

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*Ralph W. Oberste-Vorth; Aristides Mouzakitis; Bonita A. Lawrence*

MAA Press: An Imprint of the American Mathematical Society

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.

#### Reviews & Endorsements

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

# Table of Contents

## Bridge to Abstract Mathematics

- front cover cover11
- copyright page ii3
- title page iii4
- Contents vii8
- Some Notes on Notation xi12
- To the Students xiii14
- For the Professors xvii18
- I THE AXIOMATIC METHOD 122
- Introduction 324
- Statements in Mathematics 930
- Proofs in Mathematics 2950
- What is Mathematics? 2950
- Direct Proof 3051
- Contraposition and Proof by Contradiction 3354
- Proof by Induction 3758
- Proof by Complete Induction 4263
- Examples and Counterexamples 4667
- Supplemental Exercises 4869
- How to THINK about mathematics: A Summary 5273
- How to COMMUNICATE mathematics: A Summary 5273
- How to DO mathematics: A Summary 5273

- II SET THEORY 5374
- III NUMBER SYSTEMS 121142
- Algebra of Number Systems 123144
- The Natural Numbers 137158
- Summary of the Properties of the Nonnegative Integers 144165
- The Integers 145166
- Summary of the Properties of the Integers 153174
- The Rational Numbers 155176
- Summary of the Properties of the Rationals 164185
- The Real Numbers 165186
- Cantor's Reals 173194
- The Complex Numbers 181202

- IV TIME SCALES 187208
- V HINTS 209230
- Hints for (and Comments on) the Exercises 211232
- Hints for Chapter 2 211232
- Hints for Chapter 3 211232
- Hints for Chapter 4 212233
- Hints for Chapter 5 213234
- Hints for Chapter 6 214235
- Hints for Chapter 7 214235
- Hints for Chapter 8 215236
- Hints for Chapter 9 216237
- Hints for Chapter 10 216237
- Hints for Chapter 11 217238
- Hints for Chapter 12 218239
- Hints for Chapter 13 219240
- Hints for Chapter 14 220241
- Hints for Chapter 15 220241
- Hints for Chapter 16 220241

- Bibliography 223244
- Index 225246
- About the Authors 231252