An Introduction to Writing Mathematical Proofs

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 sequences, continuity, and 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.