Introduction

DM-I is an introduction to set theory, mathematical logic, and other foundational material important throughout mathematics. This section of the course will be held in English (a Slovenian section is also available). Topics include: basic set theory; relations, equivalence relations, and orders; functions; mathematical logic, truth tables, deduction, mathematical induction; and countable/uncountable cardinalities.

Prerequisite

I'll assume that you have a reasonably strong high school mathematics background.

Textbook and literature

The textbook for this course will be Transition to higher mathematics, by Bob Dumas and John McCarthy, 2nd edition. This high-quality textbook is available for free download at http://www.math.wustl.edu/~mccarthy/SandP2.pdf . Please do not buy the 1st edition, which is not free. If you like working off a printed copy (and this can be a good idea), then I suggest taking the pdf to a copy shop and asking them to print and bind it for you.

At various times in the semester, we may supplement the textbook with handouts posted to the e-classroom.

Topics

We'll begin by formally discussing some mathematical objects that you're probably familiar with: sets, functions, and sequences. Some of this material will be very familiar, but we will probably consider it a little more deeply and abstractly than you are used to. We'll continue by introducing the theory of relations, including equivalence relations and orders.

Up to this point, we'll have operated under a high-school level of argument (of proof). We'll introduce mathematical logic, and relate formal logic to mathematical proofs. Around this time we'll discuss mathematical induction, one important and useful proof technique.

From this position of maturity, we'll be able to discuss additional topics, including (un)countable cardinalities and the axiom of choice.

Exams and grading

Homework assignments will comprise 20% of your grade. The remaining 80% of your grade will be determined by exams.

There will be 2 equally weighted midterm examinations (kolokviji). The first will be roughly halfway through the semester, and the second will be around the last week of the semester, at mutually convenient times to be determined.

You may replace your midterm exam grades with your grade on the final examination (izpit). If you are highly successful on the midterm exams, you need not take the final. In accordance with university policy, there will be at least 4 opportunities to take an instance of the final. (However, you may only take the final once more after you have attained a passing grade.)

Thus, your grade will be determined by

20%(homework) + max( 40%(midterm1) + 40%(midterm2), 80%(final) )

You may bring one hand-written A4-sized sheet of paper with you to each exam.

Collaboration

I expect much of your learning to take place in working out the homework problems. While you may collaborate with other students on the homework, I expect you to have thought hard about the problems on your own first. If you do collaborate substantially, then you should so indicate on your homework paper.
Your write-up should in any case represent your own solutions, written in your own words. Copying the solutions from another student or from an internet source is a form of academic dishonesty, and will not be tolerated.

Obviously, exams will be strictly your own work.