Proofs 101: An Introduction to Formal Mathematics

✍ Scribed by Joseph Kirtland

• Year: 2020
• Tongue: English
• Leaves: 197
• Category: Library

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
0.1. TO THE STUDENT
0.2. TO THE PROFESSOR
Acknowledgments
Symbol Description
CHAPTER 1: Logic
1.1. INTRODUCTION
1.2. STATEMENTS AND LOGICAL CONNECTIVES
1.3. LOGICAL EQUIVALENCE
1.4. PREDICATES AND QUANTIFIERS
1.5. NEGATION
CHAPTER 2: Proof Techniques
2.1. INTRODUCTION
2.2. AXIOMATIC AND RIGOROUS NATURE OF MATHEMATICS
2.3. FOUNDATIONS
2.4. DIRECT PROOF
2.5. PROOF BY CONTRAPOSITIVE
2.6. PROOF BY CASES
2.7. PROOF BY CONTRADICTION
CHAPTER 3: Sets
3.1. THE CONCEPT OF A SET
3.2. SUBSETS AND SET EQUALITY
3.3. OPERATIONS ON SETS
3.4. INDEXED SETS
3.5. RUSSELL’S PARADOX
CHAPTER 4: Proof by Mathematical Induction
4.1. INTRODUCTION
4.2. THE PRINCIPLE OF MATHEMATICAL INDUCTION
4.3. PROOF BY STRONG INDUCTION
CHAPTER 5: Relations
5.1. INTRODUCTION
5.2. PROPERTIES OF RELATIONS
5.3. EQUIVALENCE RELATIONS
CHAPTER 6: Functions
6.1. INTRODUCTION
6.2. DEFINITION OF A FUNCTION
6.3. ONE-TO-ONE AND ONTO FUNCTIONS
6.4. COMPOSITION OF FUNCTIONS
6.5. INVERSE OF A FUNCTION
CHAPTER 7: Cardinality of Sets
7.1. INTRODUCTION
7.2. SETS WITH THE SAME CARDINALITY
7.3. FINITE AND INFINITE SETS
7.4. COUNTABLY INFINITE SETS
7.5. UNCOUNTABLE SETS
7.6. COMPARING CARDINALITIES
CHAPTER 8: Conclusion
CHAPTER 9: Hints and Solutions
Bibliography
Index