MAA 4211 Assignment 2

Due date: Friday, 9/26/14.

Last update made by D. Groisser Mon Sep 22 12:19 EDT 2014

You are required to do all of the problems below. You will not be required to hand them all in. I’ve indicated below which ones you do have to hand in.
Don’t make the mistake of thinking that I’m collecting only the problems I think are important.

The “due date” above is the date that your written-up problems should be handed in, but don’t
wait to get started on the assignment. You should always get started on problems as soon as we
cover the relevant material in class.

  • A: Read Rosenlicht, all of Chapter II. (You do not need to finish this before starting the problems in part C.)
  • B: Read the handout “One-to-one and onto: What you are really doing when you solve equations” posted on the Miscellaneous Handouts page. The logic of the first two pages (up to but not including the paragraph, “What this has to do with `one-to-one’ and `onto’ “) applies also to solving inequalities, such as the ones in Rosenlicht problem #5 in the book-problems below.

  • C: Rosenlicht pp. 29–31/ 4a (figure out a way to do this that does not require any division computations, works for any ordered field, and does not use a calculator), 5-11, 13, 14. Of the problems above, hand in only 5bc, 10abc, 11, 13, 14.
        Comments on some of these problems:

    • I suggest doing #11 before #10. You may find the result of #11 useful in proving the answers to one or more parts of #10.
    • In #11, I strongly suggest ignoring Rosenlicht’s hint. There is a much faster way to proceed.
    • Regarding #10c: As of Wednesday 9/17 we have not yet proved that 2 has a square root in R. We’ll prove that on Friday 9/19. If you want to get started on this problem before that, just assume that √2 exists in R.
    • In #10, replace the instruction, “giving reasons if you can” with “prove your answer.”
         If you find part 10(c) much harder than (a) or (b) (especially the “prove your answer” part), you are not going crazy! On the other hand, if you do not find it hard to prove your answer, you are probably implicitly assuming some fact we haven’t proved. If you think you have a proof, keep in mind that we have not defined what a limit is, let alone proved any properties about limits. All we have is the Least Upper Bound property of R. You need to find a way to prove your answer that does not implicitly or explicitly assume something about limits.
    • Doing #14 requires knowing there exists at least one irrational number. Once we prove that 2 has a square root in R, it will follow that at least one irrational number exists, since 2 does not have a square root in Q. To get started on this problem before Friday, you may assume that √2 exists in R.