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

Time and place:1:30PM WeH8427
Text: "Classical algebra" by Gilbert and Vanstone.
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:By appointment only.
Test Dates: First test will be held on Wedensady June 7 instead of a regular lecture.
The second test will be held on Monday June 26 instead of a regular lecture. The final will be given on Friday instead of a usual lecture.
I finished grading the test and transfered electronically the grades to the HUB. In case you want to see your test, you have to schedule 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%

The letter grade will be computed as follows: 85-100 A, 72-84 B, 60-71 C, 50-59 D and 49 or below R.
The standards of academic honesty as stated in the Student Handbook will be strictly enforced.

Assignments: A list will be handed out at the end of each lecture. 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 begin with 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 it to show 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 Gödel 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 few sections from Chapter 8.