This course serves as a transition between the calculus courses where you are primarily using mathematics to upper-level math courses where you are actually doing mathematics. We will spend a lot of time thinking about how to interpret definitions and mathematical statements, and especially how to prove them. Writing will play a significant role in this course, but this writing is of a formal type, with certain rules and accepted practices. It is not creative writing, though you need to be creative in your thinking. In many ways, this is more like poetry than prose, as it is very concerned with the form of the writing as well as its subtle meanings.
This has the following consequences for this course:
- You must know the definitions precisely. Typically, a quick restatement in your own words, while fine at an intuitive level, will lack the precision necessary for a correct use of the concept. You are responsible for all definitions from class, and you may be asked to reproduce these on exams. For example, an exam question might begin: "Give the precise definition of ..."
- You need to understand the theorems and proofs from class; it is not sufficient simply to know how to apply them. The examples we give in class are designed to help you be able to produce similar proofs on your own, so a clear understanding of each proof is crucial. You may be asked to reproduce specific proofs from class, as well as perform similar proofs of facts that you have not seen before.
- You must be very careful of the details. Simply getting "the right idea" is not enough in a mathematical proof. The real proof lies in the details, and your treatment of them will determine whether you get the proof right or not. Be particularly careful about the different "levels" involved; for example, the difference between a subset of a set and an element of a set. These are sources of trouble for many beginning students.
Some students are not sure of how much detail to go into, and where to draw the line in terms of justifying the steps of their proofs. I can offer you two pieces of advice here: first, if you had to think about it for more than a moment or two, you should explain and justify your steps; and second, if your steps involve facts or definitions that we have discussed in class, you should justify your conclusions, but if they only involve things we haven't treated in the course, you can assume the reader is aware of the fact you are using. For example, we will treat the relationship "A is a subset of B" in considerable detail, so arguments involving subsets need to be handled precisely and be fully explained. However, we will not treat the relationship "x < y" in detail, so if you know that "2x < 4", you can claim "x < 2" without further justification. I hope this will become clearer as we move through the course.