Math 300 - Transitions to Advanced Mathematics
Ruffner 337
395-2188
Office Hours: MWF 10 - 11:20, TR 10:45 - 11:20, or by appt.
Course Description: An introduction to rigorous mathematical proof with focus
on the properties of the real number system. Topics include elementary symbolic logic, mathematical induction,
algebra of sets, relations, recursion, algebraic and completeness properties of the reals. Prerequisite: MATH 262
or consent of instructor. It is strongly recommended that a student have a C or better in MATH 262. It is also
strongly recommended that a student have a C or better in this course before proceeding with future coursework.
3 credits. Offered every spring. WR
Course Objective: Students will understand various proof techniques and be able to determine when a certain technique is applicable to a given problem
Text: How to Prove it: A Structured Approach, 2nd ed. by Daniel Velleman.
Grading: You will have regular homwork, three tests, and a comprehensive Final Exam. The tests will be worth 15% each, the final will be worth 20%, and the homework will be worth 30%. The last 5% of your grade is for Culture Points. Standard grading rates apply: 90 - 100 is an A, 80 - 89 is a B, 70 - 79 is a C, 60 - 69 is a D, and anything below a 60 is failing.
| Dates | TENTATIVE SECTIONS COVERED |
|---|---|
| 1/19 - 1/21 | Deductive Reasoning, Logical Connectives, and Truth Tables, 1.1-1.2 |
| 1/24-1/28 | Sets, Operations on Sets, 1.3-1.4 |
| 1/31-2/4 | Conitionals, Biconditionals, Quantifiers, Equivalences involving Quantifiers, 1.5-2.2 |
| 2/7-2/11 | More Set Operations, Proof Strategies 2.3, 3.1 |
| 2/11-2/15 | Proofs with Negations and Conditionals, 3.2, TEST I |
| 2/14-2/18 | Proofs with Qunatifiers, Conjunctions and Biconditions, Disjunctions, 3.3-3.5 |
| 2/21-2/25 | Proofs with Disjunctions, Existence and Uniqueness Proofs, 3.5,3.6 |
| 2/28-3/4 | More Proofs, ORdered Pairs and Cartesian Pairs, 4.1 |
| 3/7-3/11 | SPRING BREAK |
| 3/14-3/18 | Relations, Ordering Relations, Closures, 4.2-4.5 |
| 3/21 - 3/25 | Closures, Equivalence Relations, Functions, 4.5-5.1 Test II |
| 3/28-4/1 | One-to-one, Onto, Inverses, Images, 5.2-5.4 |
| 4/4-4/8 | Induction and Recursion, 6.1 - 6.3, |
| 4/11-4/15 | More Induction, Equinumerous Sets, 6.5 - 7.1 |
| 4/18-4/22 | Countable and Uncountable Sets, 7.2 TEST III |
| 4/25-4/29 | Additional Material and Review |
| 5/6 | FINAL EXAM, 8 - 10:30 |
Honor Code: Every assignment you turn in for this class is covered by the Longwood College Honor Code. Please be sure that you know whether you are allowed to collaborate and which resources you are allowed to use before you start an assignment. For quizzes, tests, and the homework, you may not consult with any other person. For the quizzes and homework you may use your book and notes. The tests will be closed book. You may always consult with the instructor.
Homework: There will be several different kinds of homework. Doing this material is the best way to learn this material, so homework is very important. Traditional homework will be assigned and collected most Fridays. It is to be your own work. I might also suggest problems in the homework that are worth doing, even though they will not be collected. These you can work on using any resources you want. In addition we will do board work, hopefully at least three times during the semester. In board work, you will be given a variety of homework problems to prepare. If your name is randomly selected for a problem, you are to go up to the board and present your solution to the problem. Board work will be graded on presentation as well as correctness. I do not intend to be particularly strict enforcing the honor code when we do board work. However, I will expect you to understand the solution you are presenting. Finally, I hope to require some blogging during the semester. This will cover topics above and beyond the work we are doing in the course as well as specific proofs. I will be more explicit about this as we go.
Writing Proofs : By the end of the first third of the course your homework will involve writing proofs. Proofs should be written so that they read as correct English. Of course you are free, in fact encouraged, to use the mathematical and logical symbols that you will learn in this class as part of your written proofs. But they still must read as correct English. You will be graded not just on the mathematical and logical correctness of your proofs but on the grammatical and syntactical correctness as well. You will have some leeway early in the course, but by the end of the course it will be expected that your proofs are written correctly. This also applies to when you present your proofs in board work, when you do your blogging, and proofs you write on your tests. We will be discussing these expectations as a group throughout the semester (after we get to proofs).
Culture Club Points: Culture Points are awarded for doing a write up of out-of-class events that involve mathematics. In particular, submitting a problem of the month (and letting me have a copy of your solution) or attending a mathematical colloquium and doing a short write up will count as one culture point. Attending two math club meetings and doing a short write up of each will count as one culture point. Consult with me if you have other ideas. These assignments will be graded on how well they are written as well as mathematical correctness. If you are working on a problem of the month, you do not have to get the problem completely correct to get all of the culture points.
Attendance: You are expected to attend every class. It has been my experience that those students who do not attend class fail. You don't want to fail, so please attend class.
Advice: This class is meant to lead you from the sort of mathematics that is entirely computational into the sort of mathematics that is rigorous. More important, it is the sort of mathematics where you might not be able to see what the answer is, this is where the intuition you were born with starts to fail and you have to work to gain intuition. That intuition sometimes comes through the definition/theorem/proof structure of modern mathematics and this is the structure we are going to explore.
As always, the fact that I can do this stuff (most of the time) and you can watch me and understand what I am doing in no way implies that you can do it yourself. If you do not try to do it yourself, you will not know if you can until you turn something in. By then it might be too late. So try. You might even enjoy it. There is something strange and beautiful about the sort of mathematics that is structured pure thought.
Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics. - E.T. Bell
Pure mathematics is, in its way, the poetry of logical ideas. -A. Einstein
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. - R. Heinlein