Math 330 Introduction to Proofs

Final exam: 11-1, Monday 5/11 in S1 149.



To the final you can bring the course book, notes taken in class, any "summary sheets" you yourself make up, Homeworks, tests, and anything you print from our 330 website.
The way I would study would be to go over all the notes (I know it's a lot so you'll have to skim) and try to redo, by yourself, examples or theorems we did in class.

Things to study:

*There will be one problem to prove certain identities using only the axioms of Z, the definition of minus, and certain properties of taking negatives (like we did in chapter 1).

*There will be one induction problem.

*There will be one set theory problem. I'll give you two sets and I'll ask you to prove they are equal or prove that one is a subset of the other.

Knowing how to do the above 3 things is be the minimum amount you need to know to pass the class because they are so fundamental. To more than just skim a passing grade, you should try to know the following.

The Cantor set and its various properties.

Cardinality.

Properties of the real numbers, rational numbers and integers. For example, we used the well-ordering principle quite a bit.

The least upper bound property of the real numbers. For example, if I give you a set, can you determine its least upper bound and prove it?

Things related to divisibility. For example, besides knowing the definition of divisibility and the Fundamental Theorem of Arithmetic, we went over lots of divisibility tricks ... you don't have to know the rules, but I might give you a problem asking to prove a certain trick.

Course requirements

1) You will be actively engaged in the learning process, learn critical thinking skills, and develop problem-solving strategies. You will accomplish these goals by practicing and applying the class material.

2) Grades will be assigned based on my best judgement of how well you have learned the material of the course.

3) Please read Tips on Writing Mathematics BEFORE you turn in your first homework.

Course goals

This course has two complementary goals : (1) a rigorous development of the fundamental ideas of Calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are ``rigor" and ``proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems, propositions, lemmas, etc., and proofs.

Contents of 330: Careful discussion of the real numbers, the rational numbers and the integers, including a thorough study of induction and recursion. Countable and uncountable sets.

Contact Information

Office: LN-2224, Extension: 7-3506, Email: paul at math (put dots in) binghamton edu, Office Hours: MWF 1 - 2  and by appointment. (I'm in my office quite a bit.)

Course Materials

The Art of Proof: A Concrete Gateway to Mathematics. This textbook is published by Binghamton University and is only available from local bookstores.

Organization of Course and Weekly Schedule

We meet MWF 9:40-10:40 SW 325; T 10:05-11:30 LNG 335.

Week of:
  1. 1/26.

Homework and Tests

HOMEWORK: Homework provides the best way to reinforce the class material and to prepare for exams. I will give weekly homework assignments from the book, some of which I will collect and grade.

  1. Homework 1: Due Wed Feb 4th: Prop. 1.10 (i), (iii), (v); Prop. 1.18 (ii); Prop. 1.20.; Prop. 1.23 (ii). Please see Remarks on HW 1 for a remark on one particular question some students had trouble with.
  2. Homework 2: Due Wed Feb 11th: Do the two induction exercises I gave in class: For all natural numbers n, 2 + 2^2 + ... + 2^n = 2^(n+1) - 2 and sum_(k=1)^n k 2^k = (n-1) 2^(n+1) + 2. Also do Prop. 3.17 (ii); Prop. 3.19 (your proof CANNOT be "this follows from putting n = 0 in Cor. 3.20, which we proved in class". You should prove Prop. 3.19 from basic principles; e.g. definition of less than or other propositions that we proved.); Prop. 3.23 (iii); Finally, prove that there does NOT exist a natural number n such that 2n = 3. Please see Remarks on HW 2 for a remark on one particular question some students had trouble with.
  3. Homework 3: Due Friday March 6-th. Please see HW3 pdf file for the official HW3. (I apologize, the version on the web before was unreadable.)
  4. Homework 4: Due Fri April 17. New Version! Please see HW4 pdf file for the official HW4.
  5. Homework 5: HW5 pdf file.
TESTS:
  1. Test 1 (revised version!): Due Monday March 23-th.
  2. Test 2 : Due Thursday April 30-th.

Group Work

I strongly believe that you should work in groups: to understand lectures, do homework and study for tests. You can also do group work for homeworks, but makes sure to write on your homework who you worked with.

Grading

Remember that I don't "give" a grade; I record the grade that you earn.

(Tentative scale -- to be finalized later.)

Homework
 30%
Two tests
40%
Final (Monday, May 11, 11:00–1:00 in S1-149)
 30%


We shall use the following (approximate) grading scale. In reality, however, the final distribution is neither a straight scale nor a fixed curve. It will depend on how the class does as a whole on each of the tests and overall.

A
 90 - 100
B
 80 - 89
C
 70 - 79
D
 60 - 69
F 0 - 59

Any cases of cheating will be subject to investigation by the Academic Honesty Committee of Harpur College.

Drop Deadline

Feb 6

Withdrawal Deadline

March 27

Attendance policy and general comments

Class attendance is strongly advised. If you do not want to take this class seriously, then please do not sign up. Questions are welcome ANYTIME. 

If you have difficulties, don't wait until the last minute. Make use of my office hours, my email, my telephone, and perhaps the best of all, your own classmates. 

Disclaimer: Finally, all of the above is subject to change as the semester progresses.