Write Your Own Proofs in Set Theory and Discrete Mathematics is a comprehensive introduction to set theory, proof techniques, and related concepts. Primarily, the book is intended as a text for an "introduction to proof" course and as such treats a long list of topics in great detail. These topics include propositional logic, quantifiers, sets, functions, relations, mathematical induction, combinatorics, and more. Throughout each section, the emphasis is on proof, and the book contains a vast number of proof exercises.
In one of the forewords to the book (there are four), the authors give an insightful "pep talk" to the prospective student. The authors detail various methods for students to avoid becoming frustrated with the proof process. These include mastering the definitions and notation, not being afraid to ask questions, and letting a problem "sit," allowing your unconscious to continue to tackle the problem. The authors have clearly spent a lot of time examining their own thinking when it comes to proofs as well as helping students tackle their early proof experiences.
The vast majority of the books in this genre begin by establishing abstract rules for truth tables, symbolic logic, and so on. It is a little disappointing that this book follows the same tradition, given the authors' obvious insight into their students' proof experiences. It would have been nice to see more exposition in the beginning chapters to relate proofs and logic to natural language and reasoning. Instead, the authors present a dizzying array of symbolic information to the students, including nested quantifiers just ten short pages in. While the authors do an admirable job of trying to translate the symbolic language into English, there is precious little motivation to help students understand why all of this is necessary.
Overall, Write Your Own Proofs covers a large amount of material, including most significant proof techniques. Students using this book will be exposed to many abstract ideas that would be revisited in later math courses, and as such makes a good preparation for those courses. However, students with weaker backgrounds will have trouble understanding early on why the proofs are constructed the way they are, or why these ideas are relevant to the "real world."
James Hamblin is an Assistant Professor of Mathematics at Shippensburg University of Pennsylvania. His mathematical interests include origami, quilting, voting theory, and pretty much anything else he can get undergraduates interested in.