Download Full Text (6.1 MB)

Download Version 3 and Version 2.1 (44 KB)

Download Note to Students (62 KB)

Download Preface (106 KB)

Download Introduction to Writing Proofs in Mathematics (329 KB)

Download Logical Reasoning (460 KB)

Download Constructing and Writing Proofs in Mathematics (827 KB)

Download Mathematical Induction (458 KB)

Download Set Theory (636 KB)

Download Functions (829 KB)

Download Equivalence Relations (523 KB)

Download Topics in Number Theory (397 KB)

Download Finite and Infinite Sets (423 KB)

Download Appendix A: Guidelines for Writing Mathematical Proofs (88 KB)

Download Appendix B: Answers for the Progress Checks (409 KB)

Download Appendix C: Answers and Hints for Selected Exercises (456 KB)

Download Appendix D: List of Symbols (70 KB)

Download Index (89 KB)


Mathematical Reasoning: Writing and Proof is a text for the first college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. Version 3 of this book is almost identical to Version 2.1. The main change is that the preview activities in Version 2.1 have been renamed to beginning activities in Version 3. This was done to emphasize that these activities are meant to be completed before starting the rest of the section and are not just a short preview of what is to come in the rest of the section.

The primary goals of the text are to help students:

  • Develop logical thinking skills;
  • 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, which includes improving writing techniques, reading comprehension, and oral communication in mathematics;
  • better understand the nature of mathematics and its language.
  • Another important goal of this text is to provide students with material that will be needed for their further study of mathematics.

Important features of the book include: Emphasis on writing in mathematics; instruction in the process of constructing proofs; and an emphasis on active learning.

Publication Date



Grand Valley State University Libraries



Creative Commons License

Creative Commons Attribution-Noncommercial-Share Alike 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.


proofs course, transition course, mathematical writing, proof writing




Screencasts designed for use with the book can be found on YouTube.

Note to Instructors: Please contact Ted Sundstrom at for information about instructor resources such as solutions to the preview activities and solutions to the exercises. In your email, please include the name of your institution (school, college, or university), the course for which you are considering using the text, and a link to a website that can be used to verify your position at your institution.

This book has been reviewed by the American Institute of Mathematics Open Textbook Initiative.

Mathematical Reasoning Writing and Proof, Version 3

Included in

Mathematics Commons