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An Introduction to Mathematical Proofs

Nicholas A. Loehr
CRC Press
Publication Date: 
Number of Pages: 
[Reviewed by
Duane Graysay
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This volume is a textbook primarily suited for mathematics majors in mid-level major courses intended to introduce students to rigorous proof and argumentation in abstract mathematics courses. This book would be a strong choice for such a course: Those who teach introduction-to-proof courses will want to review this book and consider it as a potential course text and as a resource.
Similar to other texts written to introduce mathematics students to proof and argumentation in abstract mathematics, Loehr anchors the text with introductory material related to fundamentals of logic, set theory, number theory, relations, and functions. None of the contents would serve to replace a full course in that topic area, but each provides an appropriately rigorous introduction to essential definitions and rigorously established theorems.
Throughout the book, Loehr includes several features that are intended to support students. One notable feature is Loehr’s sections on “useful denials”, which he uses effectively as a recurring reference across the text. A second notable feature is Loehr’s inclusion of annotated proofs that describe some of the reasoning that lies behind the logical and mathematical content of the proof. Loehr alludes in the preface to the fact that the cognitive and rational processes that result in the construction of a proof are not typically visible in a published proof. However, Loehr’s annotated proofs across the text provide opportunities for relative novices to gain insight into the “behind-the-scenes” work of proof. These are valuable inclusions in a textbook designed for students who are still developing their capacity for proof. Loehr also includes multiple “optional” sections that provide rigorous extensions of each chapter which, while not essential (according to Loehr), are nevertheless interesting and valuable additions to the text.
The book will be a valuable addition to the shelf for any instructor who teaches intro-to-proof courses, either as a primary text or as a personal reference.


Dr. Duane Graysay is an Assistant Professor of Mathematics Education at Syracuse University. Dr. Graysay teaches methods courses for secondary pre-service mathematics teachers and undergraduate mathematics courses for majors and non-majors.