What You Can Learn from How to Prove It
A How to Prove It Guide by the University of Dating
How to Prove It: A Structured Approach (3rd Edition) is a textbook designed for systematically learning mathematical proof techniques. The author, Daniel J. Velleman, provides detailed explanations of mathematical logic and proof strategies, starting from the fundamentals of logic and progressing through to more complex mathematical proofs. Below is a detailed breakdown of each chapter, explained in Japanese.
Preface to the Third Edition
In the preface of the third edition, the author emphasizes the importance of learning mathematical proof techniques using this book. In addition to providing a foundation for understanding the concepts of proof and mathematical thinking, the book is structured to be easily accessible for beginners.
Introduction
The introduction explains the critical role that proofs play in mathematics. Proofs are not merely the results of calculations; they provide the logical framework that supports mathematical theories.
1. Sentential Logic
1.1 Deductive Reasoning and Logical Connectives
This section introduces the basics of sentential logic, including propositions and logical connectives (AND, OR, NOT, IF...THEN, etc.). A proposition is a statement that can be either true or false, and logical connectives allow us to form more complex propositions.
1.2 Truth Tables
Truth tables are used to visually represent how logical connectives affect propositions. By using truth tables, we can clearly determine whether a logical proposition is always true or false.
1.3 Variables and Sets
This section explains the concepts of variables and sets used within propositions. Understanding how set operations and properties are used in mathematical proofs is crucial.
1.4 Operations on Sets
As a foundation in set theory, this section covers operations such as union, intersection, and difference of sets.
1.5 The Conditional and Biconditional Connectives
This section helps readers understand the logical features of conditional (If...then) and biconditional (if and only if) propositions, which are frequently encountered during proofs.
2. Quantificational Logic
2.1 Quantifiers
This section covers universal quantifiers (For all) and existential quantifiers (There exists), which generalize propositions and are used to make broader logical statements.
2.2 Equivalences Involving Quantifiers
Here, the book explains logical equivalences involving quantifiers. This helps readers transform complex propositions into simpler forms.
2.3 More Operations on Sets
This section discusses more complex logical operations involving sets and quantifiers.
3. Proofs
3.1 Proof Strategies
This section explains basic strategies for progressing with proofs. It covers various methods such as proof by contradiction and mathematical induction.
3.2 Proofs Involving Negations and Conditionals
Here, the focus is on proving propositions that involve negations or conditional statements. These are important proof techniques in mathematical logic.
3.3 Proofs Involving Quantifiers
This section teaches how to prove propositions that involve quantifiers, particularly those of the form "For all x" or "There exists x."
3.4 Proofs Involving Conjunctions and Biconditionals
The section discusses how to prove propositions that involve logical conjunctions (AND) and biconditionals (if and only if).
3.5 Proofs Involving Disjunctions
This section covers how to prove propositions involving disjunctions (OR).
3.6 Existence and Uniqueness Proofs
Readers will learn how to prove the existence of an object and demonstrate that it is unique.
3.7 More Examples of Proofs
This section provides more examples to deepen the reader’s understanding of proof techniques.
4. Relations
4.1 Ordered Pairs and Cartesian Products
Here, the book introduces the concepts of ordered pairs and Cartesian products, which are essential for understanding relations in set theory.
4.2 Relations
This section explains the definition and properties of relations, which are fundamental for describing connections between elements of sets.
4.3 More About Relations
This section delves deeper into the more complex properties and operations of relations.
4.4 Ordering Relations
Here, the book explores the concept of ordering relations, which assign a specific order to the elements within a set.
4.5 Equivalence Relations
This section covers equivalence relations, their definition, and their properties. Understanding equivalence relations allows us to classify elements of a set.
5. Functions
5.1 Functions
This section introduces the definition of functions and their basic properties.
5.2 One-to-One and Onto Functions
The book explains the different types of functions, including injective (one-to-one) and surjective (onto) functions.
5.3 Inverses of Functions
This section explains the concept of inverse functions and their properties.
5.4 Closures
The book introduces the concept of closures in mathematics and their significance in various proofs.
5.5 Images and Inverse Images
This section covers the concepts of images and inverse images of functions, which are essential for understanding the behavior of functions.
6. Mathematical Induction
6.1 Proof by Mathematical Induction
This section explains how to use mathematical induction to prove propositions, particularly those involving integers.
6.2 More Examples
This section provides more examples of proofs using mathematical induction to help solidify understanding.
6.3 Recursion
This section introduces the concept of recursion and its connection to mathematical induction.
6.4 Strong Induction
Here, the book explores the technique of strong induction, which is a more powerful form of mathematical induction.
6.5 Closures Again
This section revisits the concept of closures, providing further details and their relevance to induction.
7. Number Theory
7.1 Greatest Common Divisors
This section covers the definition of greatest common divisors and how to calculate them.
7.2 Prime Factorization
Here, readers learn about prime factorization and its applications.
7.3 Modular Arithmetic
The book explains modular arithmetic and its properties.
7.4 Euler’s Theorem
This section covers Euler’s Theorem and its significance in number theory.
7.5 Public-Key Cryptography
The book introduces the concept of public-key cryptography and its applications.
8. Infinite Sets
8.1 Equinumerous Sets
This section covers the concept of infinite sets and what it means for two sets to have the same cardinality (equinumerous).
8.2 Countable and Uncountable Sets
The book explains the difference between countable and uncountable sets, providing a foundational understanding of infinite set theory.
8.3 The Cantor-Schroeder-Bernstein Theorem
This section discusses the proof and applications of the Cantor-Schroeder-Bernstein Theorem.
This book is a highly useful resource for anyone who wishes to systematically learn mathematical logic and proof techniques. It provides a strong foundation for understanding mathematics deeply and tackling complex proof problems.
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