Book of Proof by Richard Hammack is an introductory text covering the fundamentals of proving mathematical statements. This is the book that I used at Iowa State University for the course MATH 201, our introductory proofs course. Learning how to prove mathematical statements opens up countless opportunities for studying higher level mathematics. Understanding proof techniques is a necessity for developing an in-depth understanding of mathematical topics. Using logic and structuring complex arguments about mathematical topics can be highly rewarding and is unfortunately overlooked in the majority of math curricula for scientists and engineers. Hammack's text is the perfect option for someone seeking a gentle, yet rigorous, discussion of how to prove theorems. Perhaps more importantly, a by-product of learning this is that you gain a stronger ability to reason mathematically, which has potential far beyond the chalkboard of your math classroom.
The book is divided into four parts with different focuses. The first part covers the core ideas needed to begin proving statements such as set theory and logic. The second part discusses the simpler proof techniques relating to proving conditional statements (i.e., if then statements). Part three dives deeper into more proof techniques, namely proving non-conditional statements, mathematical induction, and dealing with set theory proofs. The fourth part is where the foundations laid in the previous parts are geared towards more advanced topics. Hammack introduces relations and functions, which leads into a preview of analysis with the book closing off with a proof of the Canter-Bernstein-Schroeder Theorem. If you’d like to follow along and view the contents as you read this, the author has made the text freely available online here.
Part one contains the first three chapters of the book: Sets, Logic, and Counting. The chapter on sets is divided into ten sections. In these, Hammack covers fundamental concepts such as the definition of a set, set builder notation, and set operations. The exercises are quite routine and should provide enough challenge for the definitions and operations to become clear, but they are not so difficult as to be opaque. There are solutions to the odd-numbered exercises at the back of the book, so for someone who is self-learning proofs, these exercises will be plenty to ensure they’re familiar with the content. Chapter two is where the groundwork is laid for identifying and writing mathematical statements as well as manipulating them with logical operations. These concepts are absolutely critical to understanding the rest of the content, and higher math in general. Thankfully, Hammack does a wonderful job providing detailed explanations of the definitions and ideas. He takes the time to show the student how mathematics can be expressed efficiently and clearly with special notation. This is where the student can get a peek at how math can be used as perhaps the most universal language. He also provides examples that illustrate the point clearly without being clouded in unnecessary complexity. The following chapter on counting is where some of the ideas from logic and set theory are applied to basic probability and combinatorics problems. In my course, we largely skipped this chapter, only completing a select few sections, but he does provide a substantial number of examples here. If you are interested in computer science or cryptography, this chapter will be especially interesting to you.
The second part is where the heart of the content lies. Four chapters cover the fundamental ideas one must master to prove basic mathematical statements of the form "if this, then that." Chapter four dicsusses the method of direct proof while five is on the contrapositive proof. The sixth chapter covers proof by contradiction. His discussion on direct proofs begins with definitions. Hammack shows how you can utilize defintions to prove basic results in number theory (such as proofs about divisibility). As he explains these ideas, he breaks things down into their most basic elements by applying truth tables and closely examining the structure of the proof. I found this to be highly illuminating as I was learning proofs. Having some sort of guide on how to structure an argument was helpful and will be something you rely on throughout your study of higher mathematics. Hammack's discussion of contrapositive proof continues to use number theory concepts as an accessible medium for presenting the core ideas of structuring a mathematical argument, but he goes further by introducing modular arithmetic. Section three is one that I think clearly demonstrates the authors commitment to providing the student with clear and detailed instruction. Here, Hammack breaks down the proper ettiquete and style for proof-writing and gives examples of poor form. I read and reread this section multiple times to really make it stick, as it was that worth knowing. His chapter on proof by contradiction is just as strong as the others, using the classic example of proving that the square root of 2 is irrational by reaching a contradiction. I think this chapter is one that ought to be carefully studied as a new student, since I've noticed even advanced mathematics students can get tripped up by the details of formulating a proof by contradiction.
Part three introduces more proof methods, starting with proving non-conditional statements in Chapter 7. Here he starts with "if and only if" statements. He also discusses dealing with proofs of existence by again using number theory. Chapter 8 is a very important one since it is on how to prove things about sets. As an example, when Hammack covers how to prove a set is a subset of another, he breaks down the process into clear steps for both the direct approach as well as the contrapositive way.