Last updated Wed Sep 19 18:56 EDT 2018
Due-date: Monday 9/24/18
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. 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: Rosenlicht Chap. II/ 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 these, hand in only 5c, 10ab, 11, 13, 14. In each part of 10, some points will be allotted for getting/guessing the correct numerical answer, and some points for proving that that's the answer. Read the comments below before beginning these problems.
Comments on some of these problems:
- Before doing #5, 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 \(\mbox{`one-to-one'}\) and \(\mbox{`onto'}\)") applies also to solving inequalities, such as the ones in Rosenlicht problem #5.
- In 5–7, all that's really used about that R is that it's an ordered field. (In #s 6 & 7 the LUB property has no relevance. In #5, the LUB property might have entered if some numbers had been chosen differently, but it doesn't enter for the inequalities that are actually given.) In #8, the analogous construction works with R replaced by any other ordered field, but the new field constructed would not then be called "the complex number system".
- Do #11 before #10. You may find the result of #11 useful (indispensible, in fact), in proving the answers to one part of #10.
- In #11, I strongly suggest ignoring Rosenlicht's hint. It's much faster to use an approach that's analogous the proof given in class that \({\bf N}\) is not bounded from above.
- In #10, replace the instruction, "giving reasons if you can" with "prove your answer." The hardest problem on this assignment is proving the answer to 10(c). I generally grade 10(c) as a 10-point problem, with four or five points for getting the right numerical answer, and five or six points for proving that your answer is correct. I suggest doing problems 13 and 14 first, not because they'll help you with 10(c) (they won't), but just so that getting hung up on 10(c) doesn't leave you with no time to do 13 and 14.
- If you find part 10(c) much harder than (a) or (b) (especially the "prove your answer" part), you are not going crazy! In fact, if you do not find it hard to prove your answer, you're 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 of limits. All we have is the Least Upper Bound property of \({\bf R}\). You need to find a way to prove your answer that does not implicitly or explicitly assume something about limits. (That doesn't mean that you can't use your prior experience with limits to help you guess the l.u.b. in this problem; in fact, I encourage you to do so. But proving that your guess is correct can't use the word "limit", or associated notation, at this stage of the course. We're still in Chapter II; limits aren't defined till Chapter IV. Once we cover limits, you'll be allowed to use the limit-related theorems we prove—which would make proving the answer to 10(c) much easier—but not before then.)
- Doing #14 requires knowing that there exists at least one irrational number. Now that we've proved that every positive real number (in particular, 2) has a square root in \({\bf R}\), it follows that at least one irrational number exists, since 2 has no square root in \({\bf Q}\). (You are allowed to assume the latter fact. If you've never seen it proved, let me know. The proof used to be taught in high school, but I don't know whether it still is. It's something all math majors should see before they finish college.)
- B: Click here for non-book problems. Of these, hand in only B4, B5.
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