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): DedekindPeano 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) DedekindPeano vs Peano Arithmetic  shows Exercise 3.2.5 of Lakins is misguided (PA and DedekindPeano 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 wellordering 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  ZemeloFrankel
Choice (ZFC) Axioms (the official we use in modern
mathematics) VSauce video on BanachTarski (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): CantorBernsteinSchroder  
Lecture 33 videos (online): Pidgeonhole Principle  