In this set of problems, the students are asked to plunge into writing proofs. For most students, this will be very difficult. The problems here are designed to be closely related to proofs that are presented in supplement 3, so that students can model the proofs after the examples.

As stated in my email, this is essentially identical to problem set 3 from last semester, so that students who are retaking the course or who have access to materials from last semester may have an unfair advantage. I'm not sure what to do about this; I tried changing the problems but the ones I have are carefully chosen to fit with supplement 3, and my attempts to modify them didn't work.

The assignment is on the long side; you may choose to cut some problems. I would probably choose from among the last three problems to cut.

Problem 1 is similar to the theorem proved on page 5.

Problem 2 is similar to the example on the bottom of page 10.

Problem 3 concerns a common error that I've seen in proofs. Students are asked to find this error in a proof.

Problem 4: The correct part of the false proof in problem 3 provides a useful model for this proof.

Problem 5 is a straightforward generalization of the example at the bottom of page 8.

Problem 6 (which may be the hardest problem here) is a straightforward generalization of the example at the bottom of page 9.

Problem 7 is not similar to any example in the supplement, but it is trivial once you see it; checking it for small values of n reveals the triviality. It is a nice elementary example of a ``For all x, there exists y'' type of proof.

Problem 8 is intended to give students practice ``experimenting'' with mathematics in order to find out what's true.