Mathematical Reasoning: Writing and Proof, Version 2.1
Ted Sundstrom, Grand Valley State University
Pub Date: 2014
ISBN 13: 978-1-4921038-5-1
Publisher: Grand Valley State University
Conditions of Use
This text addresses most major components of proof methods, with suitable exploration exercises for the mathematical novice/undergraduate major which read more
This text addresses most major components of proof methods, with suitable exploration exercises for the mathematical novice/undergraduate major which do not overly burden the reader with prerequisite knowledge.
The book has accurate and only contains minimal typographical errors. The mathematics in the book is correct.
This text will be relevant for a long while. Students who become math or statistics majors need to understand proof, and the basic methods used in proof and mathematical logic have not significantly changed (and will not) over time.
This is a strong point of the text. The writing is extremely clear and simple, making it easy for the undergraduate reader to follow where ,any other books fail. The examples lead the reader gently towards an understanding of logic and proof. Especially good are the sections where the author clarifies how to write a proof for your audience.
Excellent. There are simply no problems with the consistency of the mathematical work or exposition.
This can generally be done, although it takes a bit of work. Mathematics courses are often linear in this way. Despite this, with a little extra effort by an instructor, most sections can be separated. For example, it is difficult to speak of correspondences without the notion of a function, but an instructor can simply introduce the function definition to address correspondences without covering the entire chapter on functions. (By the way, one of the topics covered is “modular” arithmetic, so I am inclined to say that those parts are quite modular!)
The presentation is clear and allows the instructor to develop a natural flow to a course.
There are no issues with the interface. It is an easy-to-read pdf, of small size.
The grammar is mostly good, with only a minor error or two: so minor that it is easy to not notice them.
This book is not culturally insensitive. It is simply mathematics, and doesn’t include any offensive content in the main chapters or exercises. I am rating this a 3 as it is therefore neutral in this regard.
This book is very useful either as a primary course text for an Introduction to Proof course, or as a supplementary text in a course in philosophical logic or mathematics content course at roughly the level of Calculus I or beyond.
Table of Contents
- Note to Students
- Introduction to Writing Proofs in Mathematics
- Logical Reasoning
- Constructing and Writing Proofs in Mathematics
- Mathematical Induction
- Set Theory
- Equivalence Relations
- Topics in Number Theory
- Finite and Infinite Sets
- Appendix A: Guidelines for Writing Mathematical Proofs
- Appendix B: Answers for the Progress Checks
- Appendix C: Answers and Hints for Selected Exercises
- Appendix D: List of Symbols
About the Book
Mathematical Reasoning: Writing and Proof is designed to be a text for the ?rst course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are to help students:
- Develop logical thinking skills and to develop the ability to think more abstractly in a proof oriented setting.
- Develop the ability to construct and write mathematical proofs using standard methods of mathematical proof including direct proofs, proof by contradiction, mathematical induction, case analysis, and counterexamples.
- Develop the ability to read and understand written mathematical proofs.
- Develop talents for creative thinking and problem solving.
- Improve their quality of communication in mathematics. This includes improving writing techniques, reading comprehension, and oral communication in mathematics.
- Better understand the nature of mathematics and its language.
This text also provides students with material that will be needed for their further study of mathematics.
About the Contributors
Ted Sundstrom, Professor of Mathematics, Grand Valley State University. PhD, (Mathematics), University of Massachusetts. Dissertation: Groups of Automorphisms of Simple Rings.