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.
Some Notes on Notation
To the Students
To Those Beginning the Journey into Proof Writing
How to Use This Text
Do the Exercises!
For the Professors
To Those Leading the Development of Proof Writing
For Students in a Broad Range of Disciplines
I. The Axiomatic Method
2. Statements in Mathematics
3. Proofs in Mathematics
II. Set Theory
4. Basic Set
6. Relations on a Set
III. Number Systems
8. Algebra of Number Systems
9. The Natural Numbers
10. The Integers
11. The Rational Numbers
12. The Real Numbers
13. Cantor’s Reals
14. The Complex Numbers
IV. Time Scales
15. Time Scales
16. The Delta Derivative
17. Hints for (and Comments on) the Exercises
About the Authors
Whatever mathematics may be as a mental activity, it is communicated as a language. Therefore, it has its specific syntax, its own technical terms, and its own conventions. Mathematics is also an exact science, which means that we are obliged to express our mathematical thoughts with high precision. A deviation from the norm may lead to a complete distortion of the intended meaning.
The purpose of this chapter is to make explicit a number of principles that pertain to mathematical logic. This branch of mathematics formalizes the principles of mathematical reasoning, principles that permeate mathematical thinking, be it consciously or subconsciously.
In mathematics, we assert the truth of some statements. Other statements are to be proved or disproved. This is a fundamental dichotomy:
No mathematical statement is both true and false.
Viewed axiomatically, you can take true and false to be undefined terms; the above dichotomy can be taken as an axiom. We should clarify what is meant by a mathematical statement.
Ralph W. Oberste-Vorth was born in Brooklyn, New York and attended New York City public schools. His interests in mathematics began in junior high school and developed at Stuyvesant High School and Hunter College of the City University of New York. He met Aristides Mouzakitis at Hunter, where they became good friends while earning BA andMA degrees in mathematics. Ralph continued his studies at Cornell University where he earned his PhD in dynamical systems under the direction of John Hamal Hubbard. After oneyear positions at Yale University and the Institute for Advanced Study in Princeton, New Jersey, he moved to the University of South Florida in 1989. In 2000, Ralph approached Aristides with the idea of writing a “proofs text.” During an intense 19-day session at his home in Corfu, Greece, they wrote the first draft of this book. Several instructors at South Florida used it. Ralph moved to Marshall University as the Chairman of the Department of Mathematics in 2002. In 2009, he invited his Marshall colleague, Bonita Lawrence, to help them put the book into a publishable form. The book had been used several times at Marshall. This project was started during the summer of 2009 and completed in the summer of 2011, with input from the MAA. In August 2011, Ralph became the Chairman of the Department of Mathematics and Computer Science at Indiana State University. Ralph lives in Terre Haute, Indiana with his wife and three children.
Aristides Mouzakitis was born in the village of Avliotes on the Greek island Corfu in the Ionian Sea. He attended primary school in Avliotes and moved to Kerkyra, the main town of Corfu, to attend high school. In 1980, Aristides moved to New York City to attend Hunter College of the City University of New York. There, he earned his BA in the Special Honors Curriculum and his MA in mathematics. Aristides met Ralph Oberste-Vorth while at Hunter, where they laid the foundations for an enduring friendship. He began doctoral studies in mathematics, but Greece beckoned. In Greece, he has worked as a teacher in secondary education and as an English - Greek translator of popular mathematics books and articles. Eventually, he took on further formal studies and in 2009 he earned his doctorate in mathematics education fromthe University of Exeter in England under the direction of Paul Ernest. Aristides stays active in the Astronomical Society of Corfu and the Corfu branch of the Hellenic Mathematical Society. He lives in Kerkyra with his wife and his daughter, and enjoys reading and swimming, especially in winter time.
Bonita Lawrence is currently a Professor of Mathematics at Marshall University in Huntington, West Virginia. She was born to a military family when her father was stationed with the U.S. Army in Stuttgart, Germany. Her father retired at Ft. Sill near Lawton, Oklahoma when she was in junior high school. She received her baccalaureate degree in Mathematics Education from Cameron University in Lawton in 1979. After ten years of teaching, she returned to school to study for a Master’s degree in Mathematics at Auburn University. Upon completion of her Master’s degree in 1990, she continued her academic training at the University of Texas at Arlington, earning a Ph.D. in Mathematics in 1994. In her first teaching position after completing her Ph.D., at North Carolina Wesleyan College, she was the 1996 Professor of the Year. After a few years at small institutions, North Carolina Wesleyan College and the Beaufort Campus of the University of South Carolina, she made the move to Marshall University to expand her teaching opportunities and to work with graduate students at the Master’s level. She served as either Associate Chairman or Assistant Chairman for Graduate Studies for 10 of the 11 years under the leadership of Dr. Ralph Oberste-Vorth.
During her time at Marshall University, she has received the following research and teaching awards: Marshall University Distinguished Artists and Scholars Award—Junior Recipient for Excellence in All Fields (Spring 2002); Shirley and Marshall Reynolds Outstanding Teaching Award (Spring 2005); Marshall University Distinguished Artists and Scholars Award—Team Award for Distinguished Scholarly Activity, with one of my coauthors, Dr. Ralph Oberste-Vorth (Spring 2007); Charles E. Hedrick Outstanding Faculty Award (April 2009); and the West Virginia Professor of the Year (March 2010).
Dr. Lawrence currently is the lead researcher for the Marshall Differential Analyzer Lab, a mathematics lab that houses the Marshall Differential Analyzers. These machines, built by students of replicated Meccano components, are models of the machines that were first built in the late 1920’s to solve differential equations. The largest of the machines, a four integrator model that can run up to fourth order equations, is the only publicly accessible machine of its size and type in the country. The lab offers the opportunity for the investigation of new research ideas as well as educational experiences for students of mathematics at many levels.
This is her first book as a coauthor. She served as a reviewer for a linear algebra textbook and the solutions manual, The Keys to Linear Algebra, by Daniel Solow. Dr. Lawrence shares her life with her husband of 15 years, Dr. Clayton Brooks, a colleague in the Mathematics Department.