| Softcover ISBN: | 978-1-4704-6514-8 |
| Product Code: | TEXT/68 |
| List Price: | $85.00 |
| MAA Member Price: | $63.75 |
| AMS Member Price: | $63.75 |
| eBook ISBN: | 978-1-4704-6761-6 |
| Product Code: | TEXT/68.E |
| List Price: | $85.00 |
| MAA Member Price: | $63.75 |
| AMS Member Price: | $63.75 |
| Softcover ISBN: | 978-1-4704-6514-8 |
| eBook: ISBN: | 978-1-4704-6761-6 |
| Product Code: | TEXT/68.B |
| List Price: | $170.00 $127.50 |
| MAA Member Price: | $127.50 $95.63 |
| AMS Member Price: | $127.50 $95.63 |
| Softcover ISBN: | 978-1-4704-6514-8 |
| Product Code: | TEXT/68 |
| List Price: | $85.00 |
| MAA Member Price: | $63.75 |
| AMS Member Price: | $63.75 |
| eBook ISBN: | 978-1-4704-6761-6 |
| Product Code: | TEXT/68.E |
| List Price: | $85.00 |
| MAA Member Price: | $63.75 |
| AMS Member Price: | $63.75 |
| Softcover ISBN: | 978-1-4704-6514-8 |
| eBook ISBN: | 978-1-4704-6761-6 |
| Product Code: | TEXT/68.B |
| List Price: | $170.00 $127.50 |
| MAA Member Price: | $127.50 $95.63 |
| AMS Member Price: | $127.50 $95.63 |
-
Book DetailsAMS/MAA TextbooksVolume: 68; 2021; 334 ppMSC: Primary 00
Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics that is built around proofs and concepts.
The spirit of the book is that the basic tools of abstract mathematics are best developed in context and that creativity and imagination are at the core of mathematics. So, while the book has chapters on statements and sets and functions and induction, the bulk of the book focuses on core mathematical ideas and on developing intuition. Along with chapters on elementary combinatorics and beginning number theory, this book contains introductory chapters on real analysis, group theory, and graph theory that serve as gentle first exposures to their respective areas. The book contains hundreds of exercises, both routine and non-routine.
This book has been used for a transition to advanced mathematics courses at California State University, Northridge, as well as for a general education course on mathematical reasoning at Krea University, India.
ReadershipUndergraduate students interested in an introduction to proofs.
-
Table of Contents
-
Title page
-
Copyright
-
Contents
-
Preface
-
Chapter 1. Introduction
-
1.1. Further Exercises
-
Chapter 2. The Pigeonhole Principle
-
2.1. Pigeonhole Principle (PHP)
-
2.2. PHP Generalized Form
-
2.3. Further Exercises
-
Chapter 3. Statements
-
3.1. Statements
-
3.2. Negation of a Statement
-
3.3. Compound Statements
-
3.4. Statements Related to the Conditional
-
3.5. Remarks on the Implies Statement: Alternative Phrasing, Negations
-
3.6. Further Exercises
-
Chapter 4. Counting, Combinations
-
4.1. Fundamental Counting Principle
-
4.2. Permutations and Combinations
-
4.3. Binomial Relations and Binomial Theorem
-
4.4. Further Exercises
-
Chapter 5. Sets and Functions
-
5.1. Sets
-
5.2. Equality of Sets, Subsets, Supersets
-
5.3. New Sets From Old
-
5.4. Functions Between Sets
-
5.5. Composition of Functions, Inverses
-
5.6. Examples of Some Sets Commonly Occurring in Mathematics
-
5.7. Further Exercises
-
Chapter 6. Interlude: So, How to Prove It? An Essay
-
Chapter 7. Induction
-
7.1. Principle of Induction
-
7.2. Another Form of the Induction Principle
-
7.3. Further Exercises
-
7.4. Notes
-
Chapter 8. Cardinality of Sets
-
8.1. Finite and Infinite Sets, Countability, Uncountability
-
8.2. Cardinalities of Q and R
-
8.3. The Schröder-Bernstein Theorem
-
8.4. Cantor Set
-
8.5. Further Exericses
-
Chapter 9. Equivalence Relations
-
9.1. Relations, Equivalence Relations, Equivalence Classes
-
9.2. Examples
-
9.3. Further Exercises
-
Chapter 10. Unique Prime Factorization in the Integers
-
10.1. Notion of Divisibility
-
10.2. Greatest Common Divisor, Relative Primeness
-
10.3. Proof of Unique Prime Factorization Theorem
-
10.4. Some Consequences of the Unique Prime Factorization Theorem
-
10.5. Further Exercises
-
Chapter 11. Sequences, Series, Continuity, Limits
-
11.1. Sequences
-
11.2. Convergence
-
11.3. Continuity of Functions
-
11.4. Limits of Functions
-
11.5. Relation between limits and continuity
-
11.6. Series
-
11.7. Further Exercises
-
Chapter 12. The Completeness of R
-
12.1. Least Upper Bound Property (LUB)
-
12.2. Greatest Lower Bound Property
-
12.3. Archimedean Property
-
12.4. Monotone Convergence Theorem
-
12.5. Bolzano-Weierstrass Theorem
-
12.6. Nested Intervals Theorem
-
12.7. Cauchy sequences
-
12.8. Convergence of Series
-
12.9. 𝑛-th roots of positive real numbers
-
12.10. Further Exercises
-
Notes
-
Chapter 13. Groups and Symmetry
-
13.1. Symmetries of an equilateral triangle
-
13.2. Symmetries of a square
-
13.3. Symmetries of an 𝑛-element set
-
Groups
-
13.4. Subgroups
-
13.5. Cosets, Lagrange’s Theorem
-
13.6. Symmetry
-
13.7. Isomorphisms Between Groups
-
13.8. Further Exercises
-
Chapter 14. Graphs: An Introduction
-
14.1. Königsberg Bridge Problem and Graphs
-
14.2. Walks, Paths, Trails, Connectedness
-
14.3. Existence of Eulerian Trails and Circuits: Sufficiency
-
14.4. Further Exercises
-
Index
-
-
Additional Material
-
Reviews
-
It is well suited for self-study, but several possible selections of the chapters are suggested to set up courses of varying lengths. No special prerequisites are needed besides a basic mathematical formation and a certain desire to move to a higher mathematical level.
Adhemar François Bultheel, zbMATH
-
-
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
- Reviews
- Requests
Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics that is built around proofs and concepts.
The spirit of the book is that the basic tools of abstract mathematics are best developed in context and that creativity and imagination are at the core of mathematics. So, while the book has chapters on statements and sets and functions and induction, the bulk of the book focuses on core mathematical ideas and on developing intuition. Along with chapters on elementary combinatorics and beginning number theory, this book contains introductory chapters on real analysis, group theory, and graph theory that serve as gentle first exposures to their respective areas. The book contains hundreds of exercises, both routine and non-routine.
This book has been used for a transition to advanced mathematics courses at California State University, Northridge, as well as for a general education course on mathematical reasoning at Krea University, India.
Undergraduate students interested in an introduction to proofs.
-
Title page
-
Copyright
-
Contents
-
Preface
-
Chapter 1. Introduction
-
1.1. Further Exercises
-
Chapter 2. The Pigeonhole Principle
-
2.1. Pigeonhole Principle (PHP)
-
2.2. PHP Generalized Form
-
2.3. Further Exercises
-
Chapter 3. Statements
-
3.1. Statements
-
3.2. Negation of a Statement
-
3.3. Compound Statements
-
3.4. Statements Related to the Conditional
-
3.5. Remarks on the Implies Statement: Alternative Phrasing, Negations
-
3.6. Further Exercises
-
Chapter 4. Counting, Combinations
-
4.1. Fundamental Counting Principle
-
4.2. Permutations and Combinations
-
4.3. Binomial Relations and Binomial Theorem
-
4.4. Further Exercises
-
Chapter 5. Sets and Functions
-
5.1. Sets
-
5.2. Equality of Sets, Subsets, Supersets
-
5.3. New Sets From Old
-
5.4. Functions Between Sets
-
5.5. Composition of Functions, Inverses
-
5.6. Examples of Some Sets Commonly Occurring in Mathematics
-
5.7. Further Exercises
-
Chapter 6. Interlude: So, How to Prove It? An Essay
-
Chapter 7. Induction
-
7.1. Principle of Induction
-
7.2. Another Form of the Induction Principle
-
7.3. Further Exercises
-
7.4. Notes
-
Chapter 8. Cardinality of Sets
-
8.1. Finite and Infinite Sets, Countability, Uncountability
-
8.2. Cardinalities of Q and R
-
8.3. The Schröder-Bernstein Theorem
-
8.4. Cantor Set
-
8.5. Further Exericses
-
Chapter 9. Equivalence Relations
-
9.1. Relations, Equivalence Relations, Equivalence Classes
-
9.2. Examples
-
9.3. Further Exercises
-
Chapter 10. Unique Prime Factorization in the Integers
-
10.1. Notion of Divisibility
-
10.2. Greatest Common Divisor, Relative Primeness
-
10.3. Proof of Unique Prime Factorization Theorem
-
10.4. Some Consequences of the Unique Prime Factorization Theorem
-
10.5. Further Exercises
-
Chapter 11. Sequences, Series, Continuity, Limits
-
11.1. Sequences
-
11.2. Convergence
-
11.3. Continuity of Functions
-
11.4. Limits of Functions
-
11.5. Relation between limits and continuity
-
11.6. Series
-
11.7. Further Exercises
-
Chapter 12. The Completeness of R
-
12.1. Least Upper Bound Property (LUB)
-
12.2. Greatest Lower Bound Property
-
12.3. Archimedean Property
-
12.4. Monotone Convergence Theorem
-
12.5. Bolzano-Weierstrass Theorem
-
12.6. Nested Intervals Theorem
-
12.7. Cauchy sequences
-
12.8. Convergence of Series
-
12.9. 𝑛-th roots of positive real numbers
-
12.10. Further Exercises
-
Notes
-
Chapter 13. Groups and Symmetry
-
13.1. Symmetries of an equilateral triangle
-
13.2. Symmetries of a square
-
13.3. Symmetries of an 𝑛-element set
-
Groups
-
13.4. Subgroups
-
13.5. Cosets, Lagrange’s Theorem
-
13.6. Symmetry
-
13.7. Isomorphisms Between Groups
-
13.8. Further Exercises
-
Chapter 14. Graphs: An Introduction
-
14.1. Königsberg Bridge Problem and Graphs
-
14.2. Walks, Paths, Trails, Connectedness
-
14.3. Existence of Eulerian Trails and Circuits: Sufficiency
-
14.4. Further Exercises
-
Index
-
It is well suited for self-study, but several possible selections of the chapters are suggested to set up courses of varying lengths. No special prerequisites are needed besides a basic mathematical formation and a certain desire to move to a higher mathematical level.
Adhemar François Bultheel, zbMATH
