This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and culture.Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces.Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecture-based or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge.
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Introduction to Mathematics
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Table of contents
- Cover
- Title page
- Copyright
- Contents
- Preface
- To the Student
- To the Teacher
- Chapter 1. Sets
- Chapter 2. Sets with Structure
- Chapter 3. Logic, Briefly
- Chapter 4. Basic Proof Techniques, Briefly
- Chapter 5. Building Sets
- Chapter 6. Optional: Set Theory Axiomatics
- Chapter 7. Equivalence Relations
- Chapter 8. Functions
- Chapter 9. Advanced Proof Techniques
- Chapter 10. The Sizes of Sets
- Chapter 11. Sequences: From Numbers to Spaces
- Chapter 12. New Numbers from Completed Spaces
- Appendix A. Axioms
- Appendix B. A Summary of Proof Techniques
- Appendix C. Typography
- Bibliography
- Index
- Back Cover
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Yes, you can access Introduction to Mathematics by Scott A. Taylor in PDF and/or ePUB format, as well as other popular books in Mathematics & Mathematics General. We have over one million books available in our catalogue for you to explore.