Sets, logic, and proof are foundational topics in undergraduate mathematics. In one sense, they are basic: logic is 'natural' for mathematics undergraduates, who typically have good reasoning skills. In another sense, they are advanced: a specialist perspective is needed.How to Think About Logic, Sets, and Proof provides a friendly but mathematically serious introduction to thinking about these topics. It draws on research in mathematics education, psychology, linguistics, and philosophy to identify points where natural and mathematical thinking diverge. Part 1 contains six short introductory chapters. Chapter 1 introduces mathematical logic as a framework for validating arguments and as a solution to ambiguous communication. Chapter 2 explains how mathematical theories are structured, and how axioms, definitions, and theorems relate to sets, logic, and proof. Chapters 3, 4 and 5 introduce key ideas in sets, logic, and proof respectively. Chapter 6 discusses the syntax, semantics and pragmatics of mathematical communication.Part 2 contains eight longer chapters with deeper mathematical content. Chapter 7 discusses sets, subsets, De Morgan's laws, power sets, and infinite sets. Chapter 8 explains logical connectives, quantifiers, and negation, discussing truth tables and truth functionality. Chapter 9 discusses conditionals, drawing on research in psychology, linguistics and philosophy to explain the mathematical interpretation of 'if-then' statements. Chapter 10 discusses functions and relations, relating these foundational notions to the construction of number systems. Chapters 11, 12 and 13 discuss direct proof, indirect proof, and proof by induction respectively, relating proof structures to the preceding work on sets and logic. Chapter 14 rounds off the book, drawing together advice on mathematical communication.