What You Can Learn from Proofs: A Long-Form Math

A Proofs: A Long-Form Math Guide by the University of Dating​

"Proofs: A Long-Form Mathematics Textbook" offers a new approach to mathematical proofs, setting it apart from traditional textbooks. Written by Jay Cummings, this book is part of a series called "Long-Form Textbooks," a new style of textbook. The hallmark of this series is its departure from the conventional structure of typical textbooks — definitions, theorems, proofs, and repetition. Instead, it explains mathematics in a more approachable and intuitive manner.

Features and Approach​

This book covers a wide range of topics to help readers understand mathematical proofs, such as "intuitive proofs," "direct proofs," "set theory," "mathematical induction," "logic," "contrapositive," "proof by contradiction," "functions and relations," and many others. What stands out is the writing style. While traditional textbooks often present proofs in a concise manner, this book prioritizes a deeper understanding. Proofs are not necessarily concise, and "scratch work" or sketches of proofs are provided before the formal proof to guide students toward developing their own reasoning. The goal is to show the process of how students might arrive at a proof, fostering understanding from a broader perspective.

Additionally, the book is written in a relaxed, conversational tone, often incorporating humor. This helps lower the barriers to learning mathematics, allowing readers to absorb the material more naturally.

The "Long-Form" Style​

The "Long-Form" style contrasts with the traditional "sage on the stage" educational model. While typical textbooks present the teacher as the "sage on the page," with concise definitions and proofs, Cummings' approach provides deeper explanations and motivations, encouraging students to learn at their own pace. This style also reflects the author's critique of expensive and commercially-driven educational materials. Amid rising textbook prices, Cummings aims to provide high-quality materials at a more accessible price. The 500-page book is available for just $16 on Amazon, offering excellent value for students and scholars.

Expanding Mathematical Understanding and Application​

A unique feature of this book is that each chapter concludes with "exercise problems" or "open questions." Additionally, "Pro Tips" are included, sharing insights that the author wishes they had known when they first studied proofs, as well as comments on mathematical culture, study tips, and historical notes. These elements not only promote learning mathematics but also help develop a deep understanding of mathematical thinking.

Moreover, each chapter includes a 6–8 page introduction to other areas of mathematics, such as "Lame Theory," "Number Theory," "Topology," "Real Analysis," "Big Data," "Game Theory," "Order Theory," and "Group Theory." These introductions offer a great opportunity for students to explore other mathematical fields and gain a broader understanding of the scope of mathematics.

Overview of Chapters​

"Proofs: A Long-Form Mathematics Textbook" is designed as a textbook to explore mathematical proof techniques in depth. Each chapter provides intuitive understanding of mathematical theories and proof methods, offering techniques to solve concrete problems. Below is a summary of each chapter:

1. Intuitive Proofs​

This chapter introduces basic problems and approaches to help understand proofs intuitively.

• 1.1 Chessboard Problems: Problems involving chessboards. This section uses specific examples to deepen intuitive understanding of proofs.
• 1.2 Naming Results: How to name mathematical results. It explores how the naming of theorems and propositions affects their proofs.
• 1.3 The Pigeonhole Principle: The pigeonhole principle. It discusses how to apply this theory to real-world problems.
• 1.4 Bonus Examples: More intuitive proof problems to practice applying the learned theories.

2. Direct Proofs​

Direct proofs involve building proofs directly from definitions and axioms to reach the required conclusions.

• 2.1 Working From Definitions: How to construct proofs based on definitions.
• 2.2 Proofs by Cases: Using casework in proof construction.
• 2.3 Divisibility: Proofs related to divisibility of integers.
• 2.4 Greatest Common Divisors (GCD): Proofs about greatest common divisors.
• 2.5 Modular Arithmetic: Proofs in modular arithmetic.
• 2.6 Bonus Examples: Applications of direct proofs.

3. Sets​

This chapter covers basic operations and properties of sets and how to prove them.

• 3.1 Definitions: Definitions and basic properties of sets.
• 3.2 Proving $A \subseteq B$: Proofs showing that set A is a subset of set B.
• 3.3 Proving $A = B$: Proofs showing that two sets are equal.
• 3.4 Set Operations: Proofs related to set operations (union, intersection, difference, etc.).
• 3.5 Two Final Topics: More advanced topics in set theory.
• 3.6 Bonus Examples: Problems that deepen understanding of set theory.

4. Induction​

This chapter covers how to use mathematical induction for proofs.

• 4.1 Dominoes, Ladders and Chips: Intuitive understanding of induction through dominoes and ladders.
• 4.2 Examples: Specific induction problems.
• 4.3 Strong Induction: Theory and practice of strong induction.
• 4.4 Non-Examples: Situations where induction cannot be used.
• 4.5 Bonus Examples: Applications of induction.

5. Logic​

This chapter covers logical reasoning and techniques for proving logical statements.

• 5.1 Statements: Definitions and basic operations for propositions.
• 5.2 Truth Tables: How to use truth tables for logical proofs.
• 5.3 Quantifiers and Negations: The logical meaning of quantifiers and negations.
• 5.4 Proving Quantified Statements: Techniques for proving quantified statements.
• 5.5 Paradoxes: Problems that involve logical contradictions and how to resolve them.
• 5.6 Bonus Examples: Practice problems for logical proofs.

6. The Contrapositive​

This chapter focuses on proofs using the contrapositive of a statement.

• 6.1 Finding the Contrapositive of a Statement: How to find the contrapositive of a statement.
• 6.2 Proofs Using the Contrapositive: Methods for proving statements using their contrapositive.
• 6.3 Bonus Examples: Problems involving proofs using the contrapositive.

7. Contradiction​

This chapter covers proof by contradiction (indirect proofs).

• 7.1 Two Warm-Up Examples: Introduction to proof by contradiction.
• 7.2 Examples: Concrete examples of proof by contradiction.
• 7.3 The Most Famous Proof in History: Famous proofs in history (e.g., the proof of the irrationality of $\sqrt{2}$).
• 7.4 The Pythagoreans: The proof of the Pythagorean Theorem.
• 7.5 Bonus Examples: Practice problems involving proof by contradiction.

8. Functions​

This chapter covers the properties of functions and techniques for proving them.

• 8.1 Approaching Functions: Basic understanding of functions.
• 8.2 Injections, Surjections, and Bijections: Proofs about injective, surjective, and bijective functions.
• 8.3 The Composition: Proofs about function composition.
• 8.4 Invertibility: Proofs about the existence of inverse functions.
• 8.5 Bonus Examples: Proof problems related to functions.

9. Relations​

This chapter focuses on relations, particularly equivalence relations.

• 9.1 Equivalence Relations: Definitions and properties of equivalence relations.
• 9.2 Abstraction and Generalization: Techniques for abstraction and generalization.
• 9.3 Bonus Examples: Problems related to relations.

This textbook is designed to help students learn mathematical proof techniques progressively and deepen their understanding. Each chapter includes rich examples and exercises, ranging from basic concepts to advanced proof methods.

Relation to "Real Analytics"​

"Proofs: A Long-Form Mathematics Textbook" has a sister book titled "Real Analytics", which is also part of the "A Long-Form Mathematics" series. "Real Analytics" focuses on real analysis and, like "Proofs," aims to deepen students' understanding of mathematical intuition and concepts. Both books feature clear and comprehensive explanations, designed to make abstract mathematical concepts more accessible to students.

Conclusion​

"Proofs: A Long-Form Mathematics Textbook" stands out from traditional proof-focused textbooks by being more approachable and encouraging intuitive understanding. It serves as a valuable resource for students, offering insights into both mathematical learning and thinking. Additionally, its affordable price makes it an excellent and accessible resource for many students.

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