A Transition to Proof: An Introduction to Advanced Mathematics (Textbooks in Mathematics)
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A Transition to Proof: An Introduction to Advanced Mathematics describes writing proofs as a creative process. There is a lot that goes into creating a mathematical proof before writing it. Ample discussion of how to figure out the "nuts and bolts'" of the proof takes place: thought processes, scratch work and ways to attack problems. Readers will learn not just how to write mathematics but also how to do mathematics. They will then learn to communicate mathematics effectively. The text emphasizes the creativity, intuition, and correct mathematical exposition as it prepares students for courses beyond the calculus sequence. The author urges readers to work to define their mathematical voices. This is done with style tips and strict "mathematical do’s and don’ts", which are presented in eye-catching "text-boxes" throughout the text. The end result enables readers to fully understand the fundamentals of proof. Features: The text is aimed at transition courses preparing students to take analysis Promotes creativity, intuition, and accuracy in exposition The language of proof is established in the first two chapters, which cover logic and set theory Includes chapters on cardinality and introductory topology Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface 1: Symbolic Logic 1.1 Statements and Statement Forms 1.2 Conditional and Biconditional Connective 1.3 Arguments 1.4 Logical Deductions 2: Sets 2.1 Set Theory Basics 2.2 Properties of Sets 2.3 Quantified Statements 2.4 Multiple Quantifiers and Arguments with Quantifiers 3: Introduction to Proofs 3.1 What is Proof? 3.2 Direct Proofs 3.3 Direct Proofs: Set Element Method 3.4 Proof by Contrapositive and Contradiction 3.5 Proof by Cases 4: Mathematical Induction 4.1 Basics of Mathematical Induction 4.2 Strong Mathematical Induction 4.3 Applications of Induction: Number Theory 5: Relations 5.1 Mathematical Relations 5.2 Equivalence Relations 5.3 Order Relations 5.4 Congruence Modulo m Relation 6: Functions 6.1 Functions Defined 6.2 Properties of Functions 6.3 Composition and Invertibility 7: Cardinality 7.1 The Finite 7.2 The Infinite: Countable 7.3 The Infinite: Uncountable 8: Introduction to Topology 8.1 Topologies and Topological Spaces 8.2 Subspace and Product Topologies 8.3 Closed Sets and Closure 8.4 Continuous Functions Appendix A: Properties of Real Number System Appendix B: Proof Writing Tips Appendix C: Selected Solutions and Hints Bibliography Index
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