Paul Erdös once remarked that “you don’t have to believe in God, but you should believe in the Book, where he maintains the best proofs of all mathematical theorems, proofs that are elegant and perfect.” Introduction to Proof in Abstract Mathematics enables the reader to recognize the elements of an acceptable proof and takes the reader one step further by providing him with the tools necessary to create elegant and creative proofs, perhaps like the ones to which Erdös was referring.
The book is very explicit, developing its subject through very definite and formal inference rules that characterize the structure of a proof. These inference rules resemble those from formal logic and they demonstrate the mechanics behind constructing a proof. As the material in the book gets more complex, appropriate abbreviations and shortcuts are introduced, turning the tedious proof into simple and elegant one. Thus, the author guides the reader smoothly from numbered explicit steps to paragraph style proofs, where the logic is implicit.
The first three chapters of the book contain illustrative and detailed examples, which use elementary mathematics. That allows the reader to focus on the proof without the distractions of unfamiliar terms and ideas. These chapters include material on sets, functions, relations, operations, and the integers. One learns how to identify the parts of a proof (hypothesis, conclusion, etc) and how to write steps bottom up as they are dictated by the inference rules. From the very beginning, the author makes clear distinction between the formal mathematical statements and ordinary English statements, which is also crucial for that transition from formal to narrative proof.
The last three chapters are more informal. They concentrate on introductory material on sequences and continuous functions of a real variable, cardinality of sets of numbers, and axiomatically-defined algebraic structure via some simple group theory. These chapters show how to apply the proof techniques developed in the first three chapters to abstract algebra and advanced calculus. They are followed by six appendixes.
Introduction to Proof in Abstract Mathematics is unlike any other book on writing proofs that I have encountered. Its effectiveness comes from making things so explicit when showing how to construct a proof that the connection between the proof steps becomes obvious and the proof steps become inevitable rather than creative. The reader will learn how to interpret and write proofs by actually writing them. Experience is gained through the hard work of doing many “missing step” exercises. Ultimately the explicit use of steps and inference rules is replaced by implicit reference and the reader is taught to write paragraph-style proofs. In addition to the many practice exercises (with solutions at the end of the book), there are also supplementary problems.
This book is a pleasant undergraduate text on how to write and interpret proofs.
A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.