Mathematical Reasoning: Writing and Proof

Written by: Ted Sundstrom (Grand Valley University)

Dozens of colleges and universities have adopted this free, online mathematics textbook for their courses. Students from coast to coast and even in Bangladesh have been assigned this text while studying the process of constructing and writing proofs. It was written by Ted Sundstrom of Grand Valley University and there are even over 100 screencasts on YouTube to supplement the text.

Along with the text and screencasts, the website for this free textbook includes study guides and a short guide on writing mathematical proofs. This mathematical reasoning textbook is great for self-study as the author has provided an answer-key for selected problems and even indicates which questions have answers within the text.

When I last visited, the textbook was last updated on November 7, 2016 and labeled as version 2.1.

Table of Contents for Mathematical Reasoning

Supplementary Materials for the Instructor
1. Introduction to Writing Proofs in Mathematics
1.1 Statements and Conditional Statements
1.2 Constructing Direct Proofs
1.3 Chapter 1 Summary
2. Logical Reasoning
2.1 Statements and Logical Operators
2.2 Logically Equivalent Statements
2.3 Open Sentences and Sets
2.4 Quantifiers and Negations
2.5 Chapter 2 Summary
3. Constructing and Writing Proofs in Mathematics
3.1 Direct Proofs
3.2 More Methods of Proof
3.3 Proof by Contradiction
3.4 Using Cases in Proofs
3.5 The Division Algorithm and Congruence
3.6 Review of Proof Methods
3.7 Chapter 3 Summary
4. Mathematical Induction
4.1 The Principle of Mathematical Induction
4.2 Other Forms of Mathematical Induction
4.3 Induction and Recursion
4.4 Chapter 4 Summary
5. Set Theory
5.1 Sets and Operations on Sets
5.2 Proving Set Relationships
5.3 Properties of Set Operations
5.4 Cartesian Products
5.5 Indexed Families of Sets
5.6 Chapter 5 Summary
6. Functions
6.1 Introduction to Functions
6.2 More about Functions
6.3 Injections, Surjections, and Bijections
6.4 Composition of Functions
6.5 Inverse Functions
6.6 Functions Acting on Sets
6.7 Chapter 6 Summary
7. Equivalence Relations
7.1 Relations
7.2 Equivalence Relations
7.3 Equivalence Classes
7.4 Modular Arithmetic
7.5 Chapter 7 Summary
8. Topics in Number Theory
8.1 The Greatest Common Divisor
8.2 Prime Numbers and Prime Factorizations
8.3 Linear Diophantine Equations
8.4 Chapter 8 Summary
9. Finite and Infinite Sets
9.1 Finite Sets
9.2 Countable Sets
9.3 Uncountable Sets
9.4 Chapter 9 Summary
A Guidelines for Writing Mathematical Proofs
B Answers for the Progress Checks
C Answers and Hints for Selected Exercises
D List of Symbols

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