| Practice Exam 01 |
solutions |
| Exam 01 |
solutions |
| Exam 02 (Wednesday, November 15th) | solutions |
| John Voight's Fall 2012 Exams and Solutions | |
| John Voight's Fall 2011 Exams and Solutions | ` |
| Jonathan Sand's Spring 2017 Exams and Solutions | |
| Practice for Final |
| Homework
01 (.tex) |
due: Friday,
September 1st, 2017
|
|
| Homework
02 (.tex) |
due: Friday, September 8th, 2017 |
solutions |
| Homework
03 (.tex) |
due: Monday, September 18th, 2017 |
solutions
|
| Homework
04 (.tex) |
due: Monday, September 25th, 2017 corrections due: Monday, October 2nd 2017 |
solutions |
| Homework
05 (.tex) |
due: Friday, September 29th, 2017 corrections due: Friday, October 6th, 2017 |
solutions |
| Homework
06 (.tex) |
due: Friday, October 6th, 2017 corrections due: Monday, October 23rd, 2017 |
solutions |
| Homework
07 (.tex) |
due: Monday, October 16th, 2017 corrections due: Monday, November 6th |
solutions |
| Extra Credit (applied to next exam) comes from discussion at Lecture 15: 41m and 49s - |
due: Monday, November 6th, 2017 |
|
| Homework
08 (.tex) |
due: Monday, October 30th , 2017 corrections due: Friday, November 10th |
solutions |
| Homework
09 (.tex) |
due: Monday, November 6th, 2017 corrections due: Monday, November 27th |
solutions |
| Homework
10 (.tex) |
due: Monday, November 27th |
solutions |
| Homework 11: If you scored less than 8/10 on problem 3 of exam 2:
|
due: Friday, December 1st, 2017 no corrections. |
|
| Homework 12 (.tex) |
due: Monday, December 4th, 2017 |
solutions |
| Homework 13: (.tex) |
due: Friday, December 8th, 2017 | solutions |
| Lecture 1 (Monday, August 28th, 2017): Propositions and Formulae | Lakins 1.1 |
| Lecture
2 (Wednesday, August 30th, 2017): Propositional
Calculus/Boolean Algebra |
Lakins 1.1 De Morgan Biography |
| Lecture
3 (Friday, September 1st, 2017): Quantifiers |
Lakins 1.1 |
| Lecture
4 (Wednesday, September 6th, 2017): Implication |
Lakins 1.1 |
| Lecture
5 (Friday, September 8th, 2017): Axioms; Groups; Rings |
Hardest class of the semester. Lakins 5.1 pg 102 talks about binary operations briefly but not in a context that is useful for you guys at this stage. We introduce binary operations as a "rule" which allow us to talk about things like addition and multiplication on sets. Ring axioms not covered in Lakins but needed to talk about the integers rigorously.' Long Term Goal: prove that the integers are an integral domain. History of Ring Theory History of Group Theory Milestone: We can talk about equivalence relations on universes. The relations we develop here will be important for modular arithmetic in Lecture 26. |
| Lecture
6 (Monday, September 11th, 2017): Divisibility in Rings |
Not in Lakins but very important as
it allows us to build lots of examples. |
| Lecture 7 (Wednesday, September 13th,
2017): Methods of Proof |
Lakins 2.1, 2.2, 2.3 |
| Review Day (Friday, September 15th,
2017) |
|
| Lecture 8 (Monday, September 18th,
2017): Uniqueness as a tool |
End of section 2 |
| Lecture 9 (Wednesday, September 20th,
2017): Review of Homework 03 - Working with Groups and Special
Primes |
Lakins 7.2.0ish (Equivalence
relations -- as formulas, not as sets) but not 7.2.1 (Equivalence classes) |
| Lecture 10 (Friday, September 22nd,
2017): Induction |
Lakins 3.1 Milestone: we can prove this: Lakins 6.2.1 (Euclidean Algorithm): Fix m. For every n there exists some a and r with 0<r<m such that n = a*m + r. It is called an algorithm because it is division with remainder. |
| Lecture 11 (Monday, September 25th,
2017): Dedekind-Peano Axioms, More Induction |
Dedekind biography |
| Lecture 12 (Wednesday, September
27th, 2017): Introduction to Strong Induction |
Lakins 3.2 Lakins 6.3 (the fundamental theorem of arithmetic) |
| Lecture 13 (Friday, September 29th,
2017): Strong and Weak Induction Equivalence, Peano Arithmetic |
see Exercise 3.2.5 in Lakins Milestone: we can now define the integers rigorously and prove that they are an integral domain Milestone: we can now define x<y for integers rigorously |
| Lecture 14 (Monday, October 2nd,
2017): Signatures, Formulas, Structures, Theories, and Models |
Not in Lakins but needed to refine
Exercise 3.2.5 Milestone: We are now talking about what we are assuming to prove things. |
| Review Day (Friday, October 6th,
2017): |
|
| Exam Day (Wednesday, October 11th,
2017) |
Chapters 1 and 2 |
| Lecture 15: (Friday, October 13th, 2017) Dedekind-Peano vs Peano Arithmetic | shows Exercise 3.2.5 of Lakins is misguided (PA and Dedekind-Peano omitted in the book) |
| Lecture 16 (Monday, October 23rd,
2017) : Some Famous Summation Formulas |
Any Calculus I book |
| Lecture 17 (Wednesday, October 18th, 2017): Binomial Theorem and Ordering | Lakins 6.1.2 (the well-ordering principle) Partial orders are not covered in this book |
| Lecture 18 (with Puck Romback!
Friday, October 20th, 2017): Set Operations |
Lakins 4.1 Cantor biography |
| Lecture 19 (Monday, October 23rd, 2017 ): Cartesian Products and Power Sets | Lakins 4.2 History of Sets |
| Lecture 20 (Wenesday, October 25th, 2017): Infinite Intersections and the Least Upper Bound Property | Lakins 4.3 To show certain infinite intersections are empty one needs the Archimedean property. This is implied by the existence of least upper bounds and greatest lower bounds of the real numbers. To talk about why this works we are essentially discussing the construction of the real numbers and Dedekind cuts (which defined real numbers as breaking points between collections of rational numbers) Archimedes Biography |
| Lecture 21 (with Jonathan Sands! Friday, October 27th, 2017): Indexed Sets (Examples) | Lakins 4.3 |
| Lecture 22 (Monday, October 30th, 2017): Indexed Sets (Generalities) | Lakins 4.3 |
| Lecture 23 (Wednesday, November 1st, 2017): Functions | Lakins 5.1, 5.5 the history of the function concept |
| Lecture 24 (with Christelle Vincent! Friday, November 3rd, 2017): Injectivity and Surjectivity | Lakin 5.3 |
| Lecture 25 (Monday, November 6th, 2017): Quotients | Lakins 7.2 |
| Lecture 26 (Wednesday, November 8th, 2017): More Quotients | Lakins 7.2.1 (Equivalence Classes) Lakins 6.4, 6.5 (Congruences) Two constructions where equivalence classes matter: *the construction of the rationals as a quotient *Cantor's construction of the real numbers |
| Lecture 27 (Friday, November 10th, 2017): Compositions, Left Inverses, Right Inverses | Lakins: 5.2, 5.4 |
| Exam 2 |
|
| Lecture 28 (Friday, November 16th, 2017): Left/Right Inverses and the Axiom of Choice | Zemelo-Frankel
Choice (ZFC) Axioms (the official we use in modern
mathematics) VSauce video on Banach-Tarski (a strange consequence of the Axiom of Choice) The Axiom of Choice Zorn's Lemma (a statement which is equivalent to the Axiom of choice) appears everywhere in analysis and abstract algebra and is a fundamental tool for modern mathematics. Here are some notes for people who are interesting in seeing where this road goes: Keith Conrad's notes on Zorn's Lemma Some Remarks on Keith Conrad's webpage: Keith Conrad (UConn) has put together an amazing collection of notes on various topics which provide very good expositions to often difficult topics for first time readers. If you find yourself stuck on something these notes can be a great resource. |
| Lecture 29 (Monday, November 27th, 2017): Different Size Infinities (Cardinality) | History
of Infinity Book of Proof: Chapter 13 |
| Lecture 30 (Wednesday November 29th, 2017): Uncountable Infinity | Book of Proof: Chapter 13 |
| Lecture 31 (Friday, December 1st, 2017): Uncountable Infinity = Power Set of Naturals | Book of Proof: Chapter 13 |
| Lecture 32 (Monday, December 4th, 2017): Cantor-Bernstein-Schroder | |
| Lecture 33 videos (online): Pidgeonhole Principle | |