Introduction to Mathematics

✍ Scribed by Scott A. Taylor

• Publisher: American Mathematical Society
• Year: 2023
• Tongue: English
• Leaves: 445
• Series: Pure and applied undergraduate texts, volume 62
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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