Two great books to learn the foundations of proof writing are How to Prove It: A Structured Approach by Daniel Velleman and Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, and Ping Zhang.
Chapters in Velleman’s book address 7 respective topics:
1 Sentential Logic
2 Quantificational Logic
6 Mathematical Induction
7 Infinite Sets
After the first 3 chapters, you are well-equipped to start tackling proofs. Throughout the chapters, there are several exercises Velleman presents for readers to tackle, and then he provides detailed step by step explanations of the solutions to the exercises.
The chapters on relations and functions are heavier on applying proof strategies to more specialized mathematics content; they provide the tools needed to prove what constitutes an equivalence relation, for example. While not explicitly referenced, the Axiom of Completeness (a central concept in analysis of real numbers) receives some discussion via ordering relations, which give rise to least upper bounds and greatest lower bounds. The chapter on functions conceptually repeats much of the content on relations, which is appropriate since a function is one type of relation. Still, this chapter represents a chance for good practice on seeing functions as essentially mappings from one set to another set and learning the notation that helps to convey this mapping idea. Chapter 6 presents mathematical induction, an essential proof strategy to learn. Chapter 7 discusses how to prove statements involving infinite sets. If time is limited, focus on chapters 1, 2, 3, and 6; you’ll get the basics needed to dive into reading and writing proofs.
Velleman’s book is primarily focused on learning definitions and strategies that support proof writing. To get this exposure and to see how proof writing connects to specific mathematical topics, the Chartrand et al book fills a useful niche.
Chapters in the Chartrand et al book address 17 respective topics:
1 Communicating Mathematics
4 Direct Proof and Proof by Contrapositive
5 More on Direct Proof and Proof Contrapositive
6 Existence and Proof by Contradiction
7 Mathematical Induction
8 Prove or Disprove
9 Equivalence Relations
11 Cardinalities of Sets
12 Proofs in Number Theory
13 Proofs in Calculus
14 Proofs in Group Theory
15 Proofs in Ring Theory*
16 Proofs in Linear Algebra*
17 Proofs in Topology*
Velleman’s chapters 1, 2, 3, and 6 cover similar topics to the first 7 sections of the Chartrand text. The eighth section “Prove or Disprove” is my favorite section. Most times in working with mathematics, we do not know the answer in advance. This section recognizes that and coaches on tackling uncertainty.
The next sections addressing equivalence relations, functions, and cardinalities of sets all receive treatment in the Velleman text. But the remaining Chartrand sections apply all of these useful definitions and proof strategies to specific mathematics content relevant to number theory, real analysis (calculus), abstract algebra (group theory and ring theory), linear algebra, and topology. Each of these chapters is a useful survey, especially if you are pursuing coursework in these areas. Most awesomely, they bring proofs to life in real bona fide mathematics and electrify mathematics too by enabling readers to engage with the rationale behind why things are explained in the ways they are.
(*) Available online (for free) at www.aw.com/info/chartrand. Not included in the actual textbook.