This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics.
Author(s): Danilo R. Diedrichs, Stephen Lovett
Series: Textbooks in Mathematics
Edition: 1
Publisher: Chapman & Hall
Year: 2022
Language: English
Pages: 520
Tags: Proofs; Precision; Analysis; Algebra
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
I. Introduction to Proofs
1. Logic and Sets
1.1. Logic and Propositions
1.2. Sets
1.3. Logical Equivalences
1.4. Operations on Sets
1.5. Predicates and Quantifiers
1.6. Nested Quantifiers
2. Arguments and Proofs
2.1. Constructing Valid Arguments
2.2. First Proof Strategies
2.3. Proof Strategies
2.4. Generalized Unions and Intersections
3. Functions
3.1. Functions
3.2. Properties of Functions
3.3. Choice Functions; The Axiom of Choice
4. Properties of the Integers
4.1. A Definition of the Integers
4.2. Divisibility
4.3. Greatest Common Divisor; Least Common Multiple
4.4. Prime Numbers
4.5. Induction
4.6. Modular Arithmetic
5. Counting and Combinatorial Arguments
5.1. Counting Techniques
5.2. Concept of a Combinatorial Proof
5.3. Pigeonhole Principle
5.4. Countability and Cardinality
6. Relations
6.1. Relations
6.2. Partial Orders
6.3. Equivalence Relations
6.4. Quotient Sets
II. Culture, History, Reading, and Writing
7. Mathematical Culture, Vocation, and Careers
7.1. 21st Century Mathematics
7.2. Collaboration, Associations, and Conferences
7.3. Studying Upper-Level Mathematics
7.4. Mathematical Vocations
8. History and Philosophy of Mathematics
8.1. History of Mathematics Before the Scientific Revolution
8.2. Mathematics and Science
8.3. The Axiomatic Method
8.4. History of Modern Mathematics
8.5. Philosophical Issues in Mathematics
9. Reading and Researching Mathematics
9.1. Journals
9.2. Original Research Articles
9.3. Reading and Expositing Original Research Articles
9.4. Researching Primary and Secondary Sources
10. Writing and Presenting Mathematics
10.1. Mathematical Writing
10.2. Project Reports
10.3. Mathematical Typesetting
10.4. Advanced Typesetting
10.5. Oral Presentations
A. Rubric for Assessing Proofs
A.1. Logic
A.2. Understanding / Terminology
A.3. Creativity
A.4. Communication
B. Index of Theorems and Definitions from Calculus and Linear Algebra
B.1. Calculus
B.2. Linear Algebra
Bibliography
Index
