An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics

An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics addresses a critical gap in mathematics education, particularly for students transitioning from calculus to more advanced coursework. Recognizing the challenges many students face in proof-writing and mathematical logic, this textbook is designed for those with little to no prior experience. It provides a structured and supportive approach, guiding students through the intricacies of writing proofs while building a solid foundation in essential mathematical concepts. The book is organized into three comprehensive sections. The first section introduces elementary proof methods, beginning with fundamental topics such as sets and mathematical logic, and systematically developing the properties of real numbers and geometry from a proof-writing perspective. The second section delves into advanced proof methods, introducing quantifiers and techniques such as proof by induction, counterexamples, contraposition, and contradiction. Finally, the third section applies these techniques to a variety of mathematical topics, including functions, equivalence relations, countability, and a variety of algebraic activities, allowing students to synthesize their learning in meaningful ways. An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics not only equips students with essential proof-writing skills but also fosters a deeper understanding of mathematical reasoning. Each chapter features clearly defined objectives, fully worked examples, and a diverse array of exercises designed to encourage exploration and independent learning. Supplemented by an Instructors' Resources guide hosted online, this text is an invaluable companion for undergraduate students eager to master the art of writing mathematical proofs.

Table of Contents:
1. Introduction Section I: Elementary Proof Methods: Our First Bicycle 2. Sets and Notation - Introduction to basic set theory 3. Mathematical Logic - Basic logic needed to be able to be able to write proofs. 4. Properties of Real Numbers - Systematically builds up the properties of real numbers 5. Geometry Revisited - Approaches topics from high school geometry from the point of view of proof writing. Section II: Advanced Proof Methods: Bicycles with Multiple Gears 6. Quantifiers and Induction - Introduces quantifiers and the technique of proof by induction 7. The Three C's: Counterexamples, Contraposition, and Contradiction - Introduces these indirect proof methods Section III: Using Our Techniques: A Mathematical Tour de France 8. Fun with Functions and Relations - An Exploration and Opportunities for writing proofs involving functions and relations 9. An Amalgam of Algebraic Activities - Opportunities to write proofs for algebraic topics 10. Appendix 11. Index

About the Author :
Professor Thomas Bieske earned his PhD from the University of Pittsburgh, United States, in 1999. His research concerns partial differential equations and analysis in metric spaces, with a focus on sub-Riemannian spaces. Professor Bieske is currently serving as the Department of Mathematics and Statistics Chair of the Undergraduate Committee-Upper Level, focusing on the performance of mathematics and statistics majors in upper-level courses, at the University of South Florida, Tampa, United States.