21-127 Concepts of Mathematics 21-127 Concepts of Mathematics

Time and place: 1:30PM-2:30PM, PH A20

Text: "Classical algebra" by Gilbert and Vanstone, 4th edition.

Instructor: Prof. Rami Grossberg

Office: WEH 7204

Phone: x8482 (268-8482 from external lines), messages at 268 2545

email: Rami@cmu.edu

URL: www.math.cmu.edu/~rami

Office Hours: Immediately after class or make an appointment.

Evaluation: There will be two one hour tests (in class), homework assignments, and a three hour final. The final will cover all the course material. These will be weighted as follows:

• Each of the two midterm tests will make up 20%
• The final will make up 40%
• Homework assignments will make up 20%.

Since some of the test questions will be based on HW problems the relative weight of being able to do the HW correctly is much greater than 20%.

The letter grade will be computed as follows: 85-100 A, 70-84 B, 60-69 C, 50-59 D and 49 or below R.

Test Dates: The tests will be in class, scheduled instead of a regular lecture. The first midterm will be held on Friday June 7th. The dates of the second test and the final will be anounced.
The standards of academic honesty as stated in the Student Handbook will be enforced.

Assignments: Will be from the textbook, a list will be handed out at the lectures (and posted on the web). There will be two types of problems: with an asterisk and without. The problems with an asterisk will be collected every Tuesday and Thursday and will be graded. It is strongly recommended to do all assignments, and in case of difficulties you can consult me.

Goals: The main purpose of this course is to teach you to be able to understand, and to write mathematical proofs. Proofs, and not prescriptions, are the central part of mathematics, and give it the rigor and absoluteness that is not present in any of the other sciences. The language of mathematical proofs is formal logic, so the course will include a short study of formal logic. But it must be emphasized that this is the language of proofs only, and that in order to do proofs there must be some content. We will emphasize proof by induction, and use several induction techniques to derive some fundametal properties of numbers.

In the early twentieth century, an attempt was made to put mathematics on a completely rigorous footing, based on well accepted and fundamental axioms (this program was advocated by David Hilbert). A large part of modern mathematical practice stems from this program. The basic theory on which most parts of abstract mathematics rest is the theory of sets. We will discuss some of this theory, including the very important notion, discovered (or created) by Cantor, of how to compare the relative sizes of infinite sets. As a postscript, it should be added that Hilbert's ideals, in their fullest form, were shown by Godel to be unrealizable. However, this in no way diminishes the importance of the axiomatic method in modern mathematics.

Material to be covered: I plan to cover most of the non-trivial material of the first five chapters of the book, with a special emphasis on chapters 3, and 5 that are the more interesting (and the hardest) chapters in the book. If time permits we will cover little from Chapter 8.

Important: Due to several reasons at certain parts of the course (especially in the begining) the lecture will not follow the order of presentation of the material in the text book. I will also treat some of the material in a different form than the book does. You should attend all lectures and take carefull notes. In case you will miss a lecture you should try to get a copy of the notes from one of your classmates. While attending the lectures is not a formal requirement, based on my exprience if you will not attend all the lectures and attempt to do all the homework you are likely to fail the course. While I plan to post some information about the course on the web (including HW assignments), the web or email are not substitutes for attending lectures and recitations.