An Introduction to Writing Mathematical Proofs,

Edition 1 Shifting Gears from Calculus to Advanced Mathematics

Editors: By Thomas Bieske

Publication Date: 13 Jan 2026

Educator resources
Student resources

0 Reviews

Hazards

The publication contains no hazards

Conformance

PDF/UA-1
The publication contains a conformance statement that it meets the EPUB Accessibility 1.1, WCAG 2.1, Level AA standard. Please see https://bornaccessible.benetech.org/certified-publishers/ for further details of our compatibility testing.
The publication was certified on 20250710
Accessibility addendum
The certifier's credential is https://bornaccessible.benetech.org/certified-publishers/
For detailed accessibility information, see Elsevier’s website at https://www.elsevier.com/about/accessibility
Compatibility tested
For queries regarding accessibility information, contact [email protected]

Ways Of Reading

This e-publication is accessible to the full extent that the file format and types of content allow, on a specific reading device, by default, without necessarily including any additions such as textual descriptions of images or enhanced navigation.
Short alternative textual descriptions
Information-rich images are described by extended descriptions
Visualised data also available as non-graphical data
All contents of the digital publication necessary to use and understanding, including any text, images (via alternative descriptions), video (via audio description) is fully accessible via suitable audio reproduction.

Navigation

The contents of the PDF have been tagged to permit access by assistive technologies as per PDF-UA-1 standard.
Index with links to referenced entries
Page breaks included from the original print source
Elements such as headings, tables, etc. for structured navigation

Additional Accessibility Information

All (or substantially all) textual matter is arranged in a single logical reading order (including text that is visually presented as separate from the main text flow, e.g., in boxouts, captions, tables, footnotes, endnotes, citations, etc.). Non-textual content is also linked from within this logical reading order. (Purely decorative non-text content can be ignored).
The language of the text has been specified (e.g., via the HTML or XML lang attribute) to optimise text-to-speech (and other alternative renderings), both at the whole document level and, where appropriate, for individual words, phrases or passages in a different language.
For readers with color vision deficiency, use of color (e.g., in diagrams, graphics and charts, in prompts, or on buttons inviting a response) is not the sole means of graphical distinction or of conveying information
Content is enhanced with ARIA roles to optimize organization and facilitate navigation
Where interactive content is included in the product, controls are provided (e.g., for speed, pause and resume, reset) and labelled to make their use clear.
Content provides explanations for unusual words, abbreviations, acronyms, idioms, jargon in an accessible form, such as glossaries, scripted pop-ups.

WCAG v2.2

Product Content

Content includes any type of illustrations.
The primary content is text.
Content includes a significant number of actionable (clickable) web links to external content, downloadable resources, supplementary material, etc.
Content includes a significant number of actionable (clickable) cross-references, hyperlinked notes and annotations, or with other actionable links between largely textual elements (e.g., quiz/test questions, ‘choose your own ending’, etc.).
Additional text not part of main content
Content includes figures, diagrams, charts and/or graphs, including other ‘mechanical’ (i.e. non-photographic) illustrations.
Text within images
Content includes mathematical notations, formulae.

Note

This product relies on 3rd party tooling which may impact the accessibility features visible in inspection copies. All accessibility features mentioned would be present in the purchased version of the title.

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. It provides a structured and supportive approach, guiding students through the intricacies of writing proofs while building a solid foundation in essential mathematical concepts. Sections introduce elementary proof methods, beginning with fundamental topics such as sets and mathematical logic, systematically develop the properties of real numbers and geometry from a proof-writing perspective, and delve into advanced proof methods, introducing quantifiers and techniques such as proof by induction, counterexamples, contraposition, and contradiction.

Finally, the book 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. It 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.

Key Features

Introduces foundational topics in elementary proof methods, including sets, mathematical logic, properties of real numbers, and geometry to establish a strong basis for proof writing
Helps undergraduate students develop or enhance their proof-writing abilities, particularly those in STEM fields with a background in Calculus I.
Fills a critical gap in mathematics education by providing structured guidance for students transitioning to higher-level proof-oriented mathematics
Offers a wealth of resources, including clearly defined objectives, fully worked examples, and diverse exercises to encourage exploration and independent learning
Supplemented by an Instructor Resource Guide that includes writing prompts, group projects, and group brainstorming activities

About the author

By Thomas Bieske, Chair of the Undergraduate Committee-Upper Level, Department of Mathematics and Statistics, University of South Florida, Tampa., USA

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

ISBN: 9780443439247

Page Count: 122

Retail Price :

Access to teacher/student resources is available to registered users with approved inspection copies or confirmed adoptions. To review this material, please request an inspection copy.