This is the web page for section 5 of Math 311w taught by Tim Reluga in the autumn of 2018.

This course introduces students to the use of mathematics as a formal language. Using a theorem-proof framework much like that used in Euclid’s geometry textbook millenia ago, we will study elementary number theory advances from ancient times to our current technological age. Theories of modular arithmetic, set theory, formal logic, groups, and other discrete-math topics will be covered, with applications to encryption and digital information encoding. The course will include several writing assignments to help students develop their communications skills.

Course syllabus, including class data, contact information, office hours, and grading policies (subject to change)

Textbook and partial solutions courtesy of Gary Mullen.

- For 8/20 -- Read Section 1.1.
- Textbook practice: Section 1.1, problems 1-7
- Very simple online gcd practice
- Online pratice for gcd's: A
- Extra practice on the division theorem
- A longer explanation of circle-squaring

- For 8/27 -- Read Section 1.2.
- For 9/5 -- Read Section 1.3
- Practice homework: Section 1.3, problems 2,3,4,6,8
- Extra proof practice using GCD theorems reference sheet.
- Interactive Eratosthenes' sieve

- For 9/10 -- Read Section 1.4.
- Practice homework: Section 1.4, problems 1,2,3,5,6,7
- Extra problems for Section 1.4: A

- For 9/17 -- Read Section 1.5.
- Practice homework: Section 1.5, problems 1-5

- For 9/24 -- Read section 1.6.
- Practice homework: Section 1.6, problems 1,2,3,5,6,7,8.
- Extra problems
- Handout for RSA theorems

- For 9/28
- Martin Hellman's and Martin Gardner's classic articles on public-key encryption.

- 10/1 - Exam 1, in-class
- For 10/5 -- Read section 3.1
- Practice homework: Section 3.1, problems 1,2,3,4
- Online pratice: A, B, C
- More pratice: Making truth-tables, Translating truth-tables
- A handout on common logic rules.
- Use deduction to prove (p → r) ∧ (q → r) = (p ∨ q) → r .
- Use deduction to prove ((p → q) ∧ (q → r)) → (p → r) . (A solution)

- For 10/8 -- Read section 3.2
- Practice homework: Section 3.2, all problems

- For 10/12 -- Read section 2.1
- Practice homework: Section 2.1, problems 1,2,3,5,6,7,8,9.
- Extra problems A and B
- Show that the conjecture \((a \cap b) \cup c = a \cap (b \cup c)\) is true in general.

- Prove that \(A \cap (B \backslash C) = (A \backslash C ) \cap B\).

- Prove that \((A \cup B) \backslash C = (A \backslash C ) \cup (B \backslash C)\).

- For 10/17 -- Read section 2.2
- Practice homework: Section 2.2, problems 1,2,4,5,6,7,8,9.
- Extra problems A
- Using induction, prove that for any partition \(P\) of a finite set \(S\), the cardinality of \(S\) is equal to the sum of the cardinalities of the elements of \(P\).

- For 10/22 -- Read section 2.3
- Practice homework: Section 2.3, problems 1,2,3,4,6,7,9
- these extra problems, updated to include partial answers.
- Sochi medal counts and retroactive update
- Construct an adjacency matrix for Pell's relation, \(x \sim y\) if and only if \(y^2 | (x^4 - 1)\), on the positive integers up to 10 using a calculator.

- 11/2, second midterm in class
- For 11/9 -- Read section 4.3, Groups
- For 11/16 -- Read section 5.1 on subgroups and order
- Practice homework: Section 5.1, problems 1, 3, 6-10

- For 11/30 -- Read section 5.2 on Cosets, orbits, and Lagrange's theorem
- Practice homework: Section 5.2, problems 1, 2, 3

- Our final exam is offered on Tuesday, December 11th, 12:20 - 2:10 in Chemistry 102.

Information on our essays will be posted here.

Past quizes, posted with their answers.

- List of theorems on GCD's and Prime numbers
- List of theorems regarding congruence classes
- List of theorems regarding order and RSA
- List of logic identies

- (11-8) Brouwer–Hilbert controversy | Wikipedia
- (11-6) Sci-Fi Writer Greg Egan and Anonymous Math Whiz Advance Permutation Problem
- (10-12) The secret intellectual history of mathematics
- (09-19) Quantum theory seems to be incomplete or inconsistent!!!.
- (09-12) Old math movies from the College geometry project
- (09-10) Solving for why
- (09-07) Hyperuniformity discovered in prime numbers
- (08-26) this sentence is not true -- an analysis of self-reference is an essay related to logical paradoxes we will introduce.
- (8-25) Set Theory and Algebra in Computer Science A Gentle Introduction to Mathematical Modeling is a much deeper dive into some of the set-theory we introduce in this course.
- Prime and Prejudice: Primality Testing Under Adversarial Conditions
- Some students have found Socratica's videos on symmetry groups useful.
- Popularizers of mathematics
- Erica Klarreich's excellent mathematics writting
- Martin Gardner, math's best friend
- Ian Stewart, another great mathematics writter.

- Visual Explanations of some simple mathematical concepts (with atleast 1 mistake in the exponential growth example).
- Iodide Notebook is a new (better) way to exchange scientific research projects.
- Financial modelling for startups
- Multi-Armed Bandits, Conjugate Models and Bayesian Reinforcement Learning
- The Scientific Paper Is Obsolete. Here's What's Next. - The Atlantic
- New study sheds more light on what caused Millennium Bridge to wobble