| Softcover ISBN: | 978-1-4704-7188-0 |
| Product Code: | AMSTEXT/62 |
| List Price: | $89.00 |
| MAA Member Price: | $80.10 |
| AMS Member Price: | $71.20 |
| eBook ISBN: | 978-1-4704-7416-4 |
| Product Code: | AMSTEXT/62.E |
| List Price: | $85.00 |
| MAA Member Price: | $76.50 |
| AMS Member Price: | $68.00 |
| Softcover ISBN: | 978-1-4704-7188-0 |
| eBook: ISBN: | 978-1-4704-7416-4 |
| Product Code: | AMSTEXT/62.B |
| List Price: | $174.00 $131.50 |
| MAA Member Price: | $156.60 $118.35 |
| AMS Member Price: | $139.20 $105.20 |
| Softcover ISBN: | 978-1-4704-7188-0 |
| Product Code: | AMSTEXT/62 |
| List Price: | $89.00 |
| MAA Member Price: | $80.10 |
| AMS Member Price: | $71.20 |
| eBook ISBN: | 978-1-4704-7416-4 |
| Product Code: | AMSTEXT/62.E |
| List Price: | $85.00 |
| MAA Member Price: | $76.50 |
| AMS Member Price: | $68.00 |
| Softcover ISBN: | 978-1-4704-7188-0 |
| eBook ISBN: | 978-1-4704-7416-4 |
| Product Code: | AMSTEXT/62.B |
| List Price: | $174.00 $131.50 |
| MAA Member Price: | $156.60 $118.35 |
| AMS Member Price: | $139.20 $105.20 |
-
Book DetailsPure and Applied Undergraduate TextsVolume: 62; 2023; 415 ppMSC: Primary 00; 03
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.
ReadershipAppropriate for an Introduction to Proofs course and for undergraduate students interested in mathematical thinking and language.
-
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
-
-
Additional Material
-
RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Requests
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.
Appropriate for an Introduction to Proofs course and for undergraduate students interested in mathematical thinking and language.
-
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
