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. 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.