Last updated Mon Sep 7 21:38 EDT 2020
Each homework-assignment page will have a "last updated" line like the one above near the top of the page, so that you can easily tell whether there have been changes since the last time you looked.
Due-date: Friday, 9/11/20
You are required to do all of the problems and reading below (except for anything explicitly labeled "optional"). 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 are to 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: Bartle & Sherbert Section 1.1/ 10, 12, 14, 15, 16, 19, 20b, 21, 23. Of these, hand in only 10, 19 (as modified below; hand in all four parts). Notes on some of these problems:
- Remember that the notation "\(f^{-1}(\mbox{some set})\)" means inverse image of that set under the function \(f\), and does not entail an assumption that \(f\) has an inverse function. If you assume that an inverse function exists in any of the exercises above in which this notation appears, you'll entirely miss the point of the exercise.
- Problem 16 is related to the handout One-to-one and onto: What you are really doing when you solve equations in part C of this assignment. You can either do #16 first and then read the handout, or read the handout first and then do #16. However, if you do #16 before reading the handout, then review your work on it after reading the handout.
- In #19, add parts (a\('\)) and (b\('\)):
- (a\('\)): Show that, whether or not \(f\) is injective, the following is true: \(f^{-1}(f(E))\supseteq E\). (The notation "\(X\supseteq Y\)" is read "\(X\) contains \(Y\)", and means the same thing as \(Y\subseteq X\).)
- (b\('\)): Show that, whether or not \(f\) is surjective, the following is true: \(f(f^{-1}(H))\subseteq H\).
- In #20, part (a) (which is not assigned) does not make sense. See the handout Some mistakes and/or misleading items in Section 1.1 .
- B: Click for non-book problems. Of these, hand in only B3, B4.
- C: Non-book reading:
- Read the handout One-to-one and onto: What you are really doing when you solve equations.
- Read the handout Some mistakes and misleading items in Section 1.1 .
- Optional reading: the handout "Difference Between Inverse Functions and Inverse Images" posted on the Miscellaneous Handouts page. (I've put the link to this page here, rather than the link to the handout itself, just to make sure you see that there is such a page.) If you're having any difficulty understanding what inverse images are, or distinguishing them from inverse functions, you should find this handout helpful.
General homework page
Class home page