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.
Author(s): B. Sethuraman
Edition: 1
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 334
Tags: Proofs; Concepts; Sets; Functions; Induction; Combinatorics; Number Theory; Real Analysis; Group Theory; Graph Theory
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. ?-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 ?-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
