The Tools of Mathematical Reasoning (Pure and Applied Undergraduate Texts) (Pure and Applied Undergraduate Texts, 26)

This accessible textbook gives beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis. The importance of the logical structure of a mathematical statement as a framework for finding a proof of that statement, and the proper use of variables, is an early and consistent theme used throughout the book. Contents Preface Chapter 1. Language, Logic, and Proof 1.1. Language and logic Exercises 1.1 1.2. Proof Exercises 1.2 Chapter 2. Techniques of Proof 2.1. More direct proofs Exercises 2.1 2.2. Indirect proofs: Proofs by contradiction and contrapositive Exercises 2.2 2.3. Two important theorems Exercises 2.3 2.4. Proofs of statements involving mixed quantifiers Exercises 2.4 Chapter 3. Induction 3.1. Principle of Mathematical Induction Exercises 3.1 3.2. Strong induction Exercises 3.2 Chapter 4. Sets 4.1. The language of sets Exercises 4.1 4.2. Operations on sets Exercises 4.2 4.3. Arbitrary unions and intersections Exercises 4.3 4.4. Axiomatic set theory Chapter 5. Functions 5.1. Definitions Exercises 5.1 5.2. Function composition Exercises 5.2 5.3. One-to-one and onto functions Exercises 5.3 5.4. Invertible functions Exercises 5.4 5.5. Functions and sets Exercises 5.5 Chapter 6. An Introduction to Number Theory 6.1. The Division Algorithm and the Well-Ordering Principle Exercises 6.1 6.2. Greatest common divisors and the Euclidean Algorithm Exercises 6.2 6.3. Relatively prime integers and the Fundamental Theorem of Arithmetic Exercises 6.3 6.4. Congruences Exercises 6.4 6.5. Congruence classes Exercises 6.5 Chapter 7. Equivalence Relations and Partitions 7.1. Introduction Exercises 7.1 7.2. Equivalence relations Exercises 7.2 7.3. Partitions Exercises 7.3 Chapter 8. Finite and Infinite Sets 8.1. Introduction Exercises 8.1 8.2. Finite sets Exercises 8.2 8.3. Infinite sets Exercises 8.3 8.4. What next? Chapter 9. Foundations of Analysis 9.1. Introduction 9.2. The Completeness Axiom Exercises 9.2 9.3. The Archimedean Property and its consequences Exercises 9.3 9.4. What next? Appendix. Writing Mathematics Bibliography Index Cover -- Title page -- Contents -- Preface -- To Students -- Acknowledgements -- Chapter 1. Language, Logic, and Proof -- 1.1. Language and logic -- 1.2. Proof -- Chapter 2. Techniques of Proof -- 2.1. More direct proofs -- 2.2. Indirect proofs: Proofs by contradiction and contrapositive -- 2.3. Two important theorems -- 2.4. Proofs of statements involving mixed quantifiers -- Chapter 3. Induction -- 3.1. Principle of Mathematical Induction -- 3.2. Strong induction -- Chapter 4. Sets -- 4.1. The language of sets -- 4.2. Operations on sets -- 4.3. Arbitrary unions and intersections -- 4.4. Axiomatic set theory -- Chapter 5. Functions -- 5.1. Definitions -- 5.2. Function composition -- 5.3. One-to-one and onto functions -- 5.4. Invertible functions -- 5.5. Functions and sets -- Chapter 6. An Introduction to Number Theory -- 6.1. The Division Algorithm and the Well-Ordering Principle -- 6.2. Greatest common divisors and the Euclidean Algorithm -- 6.3. Relatively prime integers and the Fundamental Theoremof Arithmetic -- 6.4. Congruences -- 6.5. Congruence classes -- Chapter 7. Equivalence Relations and Partitions -- 7.1. Introduction -- 7.2. Equivalence relations -- 7.3. Partitions -- Chapter 8. Finite and Infinite Sets -- 8.1. Introduction -- 8.2. Finite sets -- 8.3. Infinite sets -- 8.4. What next? -- Chapter 9. Foundations of Analysis -- 9.1. Introduction -- 9.2. The Completeness Axiom -- 9.3. The Archimedean Property and its consequences -- 9.4. What next? -- Writing Mathematics -- Bibliography -- Index -- Back Cover Offers beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis.