Introduction to Mathematics : Number, Space, and Structure

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. Cover Title page Copyright Contents Preface Who is this book for? Acknowledgments To the Student To the Teacher Prerequisites Advice for teaching from this book Chapter 1. Sets 1.1. Sets, informally 1.2. Proving set membership 1.3. Subsets 1.4. Sets whose elements are sets 1.5. Proving set equality 1.6. Uniqueness of certain elements 1.7. Additional exercises Chapter 2. Sets with Structure 2.1. Groups 2.2. Metric spaces 2.3. Graphs 2.4. The natural numbers 2.5. Application: Symmetry groups 2.6. Appendix: Euclidean metric Chapter 3. Logic, Briefly 3.1. Statements, predicates, and quantifiers 3.2. Conjunctions and disjunctions 3.3. Negations 3.4. Implications 3.5. A remark on uniqueness 3.6. Basic exercises in logic 3.7. Russell’s paradox 3.8. Application: The halting problem Chapter 4. Basic Proof Techniques, Briefly 4.1. Direct proof 4.2. Proof by contraposition 4.3. Proof by contradiction 4.4. Existence 4.5. Uniqueness 4.6. Application: p-values and scientific reasoning 4.7. Writing well 4.8. Additional proofs Chapter 5. Building Sets 5.1. Subsets 5.2. Complements 5.3. Intersections 5.4. Unions 5.5. Power sets 5.6. Cartesian products 5.7. The persistence of structure 5.8. Application: Configuration spaces 5.9. Application: The geometric structure of data 5.10. Additional problems Chapter 6. Optional: Set Theory Axiomatics 6.1. The ZFC axioms 6.2. The controversies 6.3. The existence of a natural number system 6.4. The existence of the Cartesian product 6.5. Functions, formally Chapter 7. Equivalence Relations 7.1. Partitions 7.2. Equivalence relations 7.3. Equivalence classes 7.4. Quotient sets 7.5. Equivalence relations vs. partitions 7.6. Angle addition 7.7. Constructing the integers and rationals 7.8. Modular arithmetic 7.9. Application: Configuration spaces of unlabeled points 7.10. Additional problems Chapter 8. Functions 8.1. The definition of a function 8.2. Visualizing functions 8.3. Important functions 8.4. Extended examples 8.5. Combining and adapting functions 8.6. Being well defined 8.7. Properties of functions 8.8. Application: Affine encryption 8.9. Application: Campanology 8.10. Application: Probability functions 8.11. Application: Electrical circuits 8.12. Additional problems Chapter 9. Advanced Proof Techniques 9.1. Regular old induction 9.2. Complete induction 9.3. Well-ordering principle 9.4. Constructing sequences recursively 9.5. Other induction methods 9.6. Application: Probability 9.7. Application: Iterated function systems 9.8. Application: Paths in graphs 9.9. Additional exercises 9.10. Appendix: The well-ordering theorem Chapter 10. The Sizes of Sets 10.1. Finite sets 10.2. Infinite sets 10.3. Countable sets 10.4. Uncountable sets 10.5. Producing larger cardinalities 10.6. The Cantor–Bernstein theorem 10.7. Application: Transcendental numbers 10.8. Application: Countable sets and probability 10.9. The cardinal numbers 10.10. Application: Cardinality and symmetry 10.11. Application: Dimension and space-filling curves 10.12. Application: Infinity in the humanities Chapter 11. Sequences: From Numbers to Spaces 11.1. Subsequences 11.2. Convergent sequences 11.3. Completeness 11.4. Sequences and subsequences in R 11.5. Application: Circular billiards 11.6. Additional problems Chapter 12. New Numbers from Completed Spaces 12.1. Metric completions 12.2. The 10-adic numbers 12.3. Constructing R Appendix A. Axioms Appendix B. A Summary of Proof Techniques Appendix C. Typography Bibliography Index Back Cover