
The final will be accumulative. You can expect approximately 1/3 of the problems to test on the material from the first midterm (set theory and logic), 1/3 to test on material from the second midterm (proof techniques and number theory), and 1/3 to test on the more recent material. Here is a more detailed list of the recent material:
Modular arithmetic (Houston Ch. 29 and Ch. 31)  
Solving quadratic and linear equations arising in modular arithmetic  
Equivalence relations (Houston Ch. 31)  
The completeness axiom (Beck 8.4)  
Upper/lower bounds, max/min, sup/inf (Beck 8.4)  
Theorem 10.1 and Proposition 10.4 from Beck  
Propositions 10.10 and 10.11 from Beck  
Definition of the limit of a sequence (Beck 10.4)  
Convergence/divergence (Beck 10.4)  
Increasing, decreasing, monotone (Beck 10.4)  
Theorem 10.19, Propositions 10.21, 10.22, 10.23 from Beck  
Infinite limits 
Here is an exhaustive list of material that could show up on the final.
Remember that having the correct solution to the problem is only part of the story. You also need to express your solution coherently. In studying for the exam you should practice writing up your solutions to problems in essay format, since this is how you are expected to answer problems on the exam.
Here is a set of review problems for the final. You do not need to hand this in. Solutions.
I will be holding additional office hours on Saturday 12/7 from 13 pm and on Sunday 12/8 from 24 pm.
This midterm will test on a subset of the following:
Quantifiers (for all, there exists), and negation of quantifiers  
Proofs involving 'if and only if'  
Direct proof  
Proof by contradiction  
Proof by contrapositive  
Proof by cases  
Induction  
Irrationality and square roots  
Fundamental Theorem of Arithmetic (Theorem 25.5)  
Divisibility: divisors, gcd (Theorems 27.5, 27.9, 27.20)  
Division Lemma (Lemmas 28.1 and 28.2)  
Euclidean Algorithm (Theorem 28.3) and calculating the gcd  
Theorem 28.7  
Euclid's Lemma (Corollary 28.9, Theorem 28.12) 
Here is a set of review problems for Midterm 2. You do not need to hand this in. Solutions
The MLC will be holding a review session in Wells Hall A108 on Wednesday Nov. 6, from 7:30 pm to 9:50 pm.
Here is some additional review material. Many of these problems will be covered in the MLC review session. Warning: There is some material in these worksheets that you are NOT expected to know for the exam (see above for what you ARE expected to know). Review 1, Review 2, Review 3, Review 4.
The midterm will test on a subset of the following:
Basic set operations (union, intersection, complement, subset  
Identifying functions and their domains/codomains  
Injections, surjections, bijections  
Bijective correspondence of sets  
Countability  
Basic logic operations (or, and, implies)  
Negations  
Conditional statements (hypothesis, conclusion, sufficient/necessary conditions)  
Inverse, converse, contrapositive  
Biconditional statements 
Here is a set of review problems for Midterm 1. You do not need to hand this in. This review sheet is designed to focus your attention on what to study for the midterm. If you understand how to do all problems on this review sheet, then you should have no difficulty understanding the problems on the midterm.
12/6: I will be holding additional office hours on Saturday 12/7 from 13 pm and on Sunday 12/8 from 24 pm.
11/15: Here is the handout from class on the real number axioms. Here is a version with more details (you are not required to hand in the two exercises at the end).
10/16: Here is the Methods of Proof Handout from class today. Here are solutions.
10/14: Here are solutions to the recitation worksheet from Oct. 10: Page 1, Page 2
10/9: Here are some notes on axioms.
9/24: The rewrite of essay 1 is due on Friday, October 4. Here are some more details. Here are solutions.
9/20: The Math Learning Center in Wells Hall offers tutoring for this class at the following times: M, T, W 1:404 and 6:208:40; Th 6:208:40. Our TA, Minh, is there M, T, W 2:504:00, and also M, W 7:308:40.
9/13: Please double space your solutions to Essay 1.
9/11: The due date for Essay 1 has been extended from Monday 9/16 to Wednesday 9/18.
9/11: The assignment for Homework 5 has been modified. (Problems 4 and 5 are no longer required.)
9/9: Here is the supplement handed out today in class.
8/28: Dr. Schenker has posted some useful comments for students who are beginning to write mathematics. (Dr. Schenker was the instructor for the summer session of this course. This summer session was the very first time Math 299 was offered at MSU.)
8/28: The Math Learning Center is an excellent (and free!) resource. Note that they have certain hours dedicated specifically to Math 299.
8/28: There will be a quiz during the first recitation (8/29) of the semester. It will cover the material from the first lecture (8/28).