The Long-Form Math Textbook Series: Proofs

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This textbook is tailored for students, moving beyond the typical definition-theorem-proof format to include extensive commentary, motivation, and explanations. The proofs prioritize understanding over brevity, with many preceded by "scratch work" or sketches to provide a broader perspective and guide students in developing their own proofs. Key topics include intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions, and relations, all supported by over 200 illustrations. The writing style is relaxed and conversational, featuring moments of humor. Additionally, the text serves as an introduction to higher mathematics, with selected examples and theorems that lead into various mathematical areas, such as Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality, and group theory. Each chapter concludes with "pro-tips" that offer insights on the material, study strategies, historical context, and aspects of mathematical culture. Following the exercises, readers will find introductions to unsolved problems in mathematics. The appendices cover further proof methods, showcase particularly beautiful proofs, and provide writing advice.