The catalog description for Mathematical Reasoning (MATH 3000) is: "Introduces mathematical methods of proof using primarily the subjects of logic, set theory, number theory, and equivalence relations." The prerequisites are Calculus 2 (MATH 1920) and Linear Algebra (MATH 2010). This description is based on the ETSU 2021-22 Undergraduate Catalog. The preface of the book recommends covering 1.1-1.5, Chapter 2 (except perhaps 2.2), Chapter 3 quickly, 4.1-4.3, 5.2, 5.3, 5.5-5.8, 6.3-6.6 for a one semester course. To meet our emphasis on number theory, we emphasize Chapter 6 over Chapter 5 (the topics of Chapter 5 are covered in ETSU's "Applied Combinatorics and Problem Solving" [MATH 3340]; the topics of Chapter 6 are covered in ETSU's "Elementary Number Theory" [MATH 3120]).

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

1. Chapter 1. Logic.
2. Chapter 2. Sets.
3. Chapter 3. Functions.
4. Chapter 4. Finite and Infinite Sets.
5. Chapter 5. Combinatorics.
6. Chapter 6. Number Theory.
7. Chapter 7. Complex Numbers.

Chapter 1. Logic.

• Section 1.1. Statements, Propositions, and Theorems. Section 1.1 notes
• Section 1.2. Logical Connectives and Truth Tables. Section 1.2 notes
• Section 1.3. Conditional Statements. Section 1.3 notes
• Section 1.4. Proofs: Structures and Strategies. Section 1.4 notes
• Section 1.5. Logical Equivalence. Section 1.5 notes
• Section 1.6. Application: A Brief Introduction to Switching Circuits. Section 1.6 notes
• Study Guide 1

Chapter 2. Sets.

• Section 2.1. Fundamentals. Section 2.1 notes
• Supplement. Proofs of Theorems in Section 2.1. Beamer file of Section 2.1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.1. Print file of Section 2.1 proofs
• Section 2.2. Russell's Paradox. Section 2.2 notes
• Section 2.3. Quantifiers. Section 2.3 notes
• Section 2.4. Set Inclusion. Section 2.4 notes
• Supplement. Proofs of Theorems in Section 2.4. Beamer file of Section 2.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.4. Print file of Section 2.4 proofs
• Section 2.5. Union, Intersection, and Complement. Section 2.5 notes
• Supplement. Proofs of Theorems in Section 2.5. Beamer file of Section 2.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.5. Print file of Section 2.5 proofs
• Section 2.6. Indexed Sets. Section 2.6 notes
• Supplement. Proofs of Theorems in Section 2.6. Beamer file of Section 2.6 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.6. Print file of Section 2.6 proofs.
• Section 2.7. The Power Set. Section 2.7 notes
• Supplement. Proofs of Theorems in Section 2.7. Beamer file of Section 2.7 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.7. Print file of Section 2.7 proofs
• Section 2.8. Ordered Pairs and Cartesian Products. Section 2.8 notes
• Supplement. Proofs of Theorems in Section 2.8. Beamer file of Section 2.8 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.8. Print file of Section 2.8 proofs
• Section 2.9. Set Decomposition: Partitions and Relations. Section 2.9 notes
• Supplement. Proofs of Theorems in Section 2.9. Beamer file of Section 2.9 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.9. Print file of Section 2.9 proofs
• Section 2.10. Mathematical Induction and Recursion. Section 2.10 notes
• Supplement. Proofs of Theorems in Section 2.10. Beamer file of Section 2.10 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.10. Print file of Section 2.10 proofs
• Study Guide 2

Chapter 3. Functions.

• Section 3.1. Definitions and Examples. Section 3.1 notes
• Section 3.2. Surjections, Injections, Bijections, Sequences. Section 3.2 notes
• Supplement. Proofs of Theorems in Section 3.2. Beamer file of Section 3.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 3.2. Print file of Section 3.2 proofs
• Section 3.3. Composition of Functions. Section 3.3 notes
• Supplement. Proofs of Theorems in Section 3.3. Beamer file of Section 3.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 3.3. Print file of Section 3.3 proofs
• Study Guide 3

Chapter 4. Finite and Infinite Sets.

• Section 4.1. Cardinality; Fundamental Counting Principles. Section 4.1 notes
• Supplement. Proofs of Theorems in Section 4.1. Beamer file of Section 4.1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.1. Print file of Section 4.1 proofs
• Section 4.2. Comparing Sets, Finite or Infinite. Section 4.2 notes (This section includes a biography of Georg Cantor.)
• Supplement. Proofs of Theorems in Section 4.2. Beamer file of Section 4.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.2. Print file of Section 4.2 proofs
• Section 4.3. Countable and Uncountable Sets. Section 4.3 notes
• Supplement. Proofs of Theorems in Section 4.3. Beamer file of Section 4.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.3. Print file of Section 4.3 proofs
• Section 4.4. More on Infinity. Section 4.4 notes
• Supplement. Proofs of Theorems in Section 4.4. Beamer file of Section 4.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.4. Print file of Section 4.4 proofs
• Section 4.5. Languages and Finite Automata. Section 4.5 notes
• Supplement. Proofs of Theorems in Section 4.5. Beamer file of Section 4.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.5. Print file of Section 4.5 proofs
• Study Guide 4

Chapter 5. Combinatorics.

• Section 5.1. Combinatorial Problems.
• Section 5.2. The Addition and Product Rules (review).
• Section 5.3. Introduction to Permutations.
• Section 5.4. Permutations and Geometric Symmetry.
• Section 5.5. Decomposition into Cycles.
• Section 5.6. The Order of a Permutation; A Card-Shuffling Example.
• Section 5.7. Odd and Even Permutations; Applications to Configurations.
• Section 5.8. Binomial and Multinomial Coefficients.
• Section 5.9. Graphs.
• Study Guide 5.

Chapter 6. Number Theory.

• Section 6.1. Operations. Section 6.1 notes
• Supplement. Proofs of Theorems in Section 6.1. Beamer file of Section 6.1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.1. Print file of Section 6.1 proofs
• Section 6.2. The Integers: Operations and Order. Section 6.2 notes
• Supplement. Proofs of Theorems in Section 6.2. Beamer file of Section 6.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.2. Print file of Section 6.2 proofs
• Section 6.3. Divisibility: The Fundamental Theorem of Arithmetic. Section 6.3 notes
• Supplement. Proofs of Theorems in Section 6.3. Beamer file of Section 6.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.3. Print file of Section 6.3 proofs
• Section 6.4. Congruence; Divisibility Tests. Section 6.4 notes
• Supplement. Proofs of Theorems in Section 6.4. Beamer file of Section 6.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.4. Print file of Section 6.4 proofs
• Section 6.5. Introduction to Euler's Function. Section 6.5 notes
• Supplement. Proofs of Theorems in Section 6.5. Beamer file of Section 6.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.5. Print file of Section 6.5 proofs
• Section 6.6. The Inclusion-Exclusion Principle and Euler's Function. Section 6.6 notes
• Supplement. Proofs of Theorems in Section 6.6. Beamer file of Section 6.6 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.6. Print file of Section 6.6 proofs
• Section 6.7. More on Prime Numbers. Section 6.7 notes (This section includes information on Fermat numbers.)
• Supplement. Proofs of Theorems in Section 6.7. Beamer file of Section 6.7 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.7. Print file of Section 6.7 proofs
• Section 6.8. Primitive Roots and Card Shuffling. Section 6.8 notes
• Supplement. Proofs of Theorems in Section 6.8. Beamer file of Section 6.8 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.8. Print file of Section 6.8 proofs
• Section 6.9. Perfect Numbers, Mersenne Primes, Arithmetic Functions. Section 6.9 notes
• Supplement. Proofs of Theorems in Section 6.9. Beamer file of Section 6.9 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.9. Print file of Section 6.9 proofs
• Section 6.10. Number Theory and Cryptography: A Brief Glimpse. Section 6.10 notes
• Study Guide 6

Chapter 7. Complex Numbers.

• Section 7.1. Complex Numbers.
• Section 7.2. The Gaussian Integers (partial). Section 7.2 notes
• Supplement. Proofs of Theorems in Section 7.2. Beamer file of Section 7.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 7.2. Print file of Section 7.2 proofs
• Study Guide 7.

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