Last updated Tue Sep 7 14:10 EDT 2021
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Due-date: Friday, 9/10/21
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.
Notation such as "Abbott 1.2/ 5,6" means "Abbott exercises 1.2.5, 1.2.6."
- A: Abbott 1.2/ 3, 4, 7–12. (See notes below before starting; I've modified a couple of these problems.)
Of the Abbott problems, hand in only 1.2/ 4, 7, 9, 10, 11a, with these reductions: (i) For 7a, hand in only the "find \(f(A)\) and \(f(B)\)" portion. (ii) For 9a, hand in only the "find \(f^{-1}(A)\) and \(f^{-1}(B)\)" portion.
Notes on some of these problems:
- 1.2/7:
- In the definition at the beginning of the problem, the domain and codomain of \( f \) can be arbitrary sets. You should assume this generality throughout the problem, except in part (a). In particular, in part (c), take the domain and codomain to be arbitrary sets, not \(\bf R\) (and not necessarily the same set).
- For a course at this level, the first sentence of part (a) (and of 1.2/9(a)) is poorly worded. Better would have been "Define \(f:{\bf R}\to {\bf R}\) by \(f(x)=x^2\)," or "Let \(f(x)=x^2\), where we take the domain and codomain of \(f\) both to be \(\bf R\)," etc. The awkardness of the extra words is annoying, but important right now.
As stated in class, to specify a function, you need to specify a domain, a codomain, and a "rule" (take "rule" here as short-hand word for "assignment of a unique element of the codomain to each element of the domain;" we don't mean "rule" in the restricted sense of a recipe, algebraic formula, etc.). The rule alone isn't enough. If I were to see this on work handed in to me, I would not give it full credit, whether the writer were a student, another professor, or my clone. (That doesn't mean that you won't see me make slip-ups that are as bad as or worse than this one! Perfect I ain't.)
An introductory sentence like "Let \(f(x)=x^2\)" is common in Calculus 1, where the codomain of every function is \({\bf R}\) and the domain of every function is a subset of \({\bf R}\). It's also not uncommon, even in higher-level classes or among mathematicians speaking to each other, to introduce a function with such a sentence, when the writer/speaker and listener have already established context that makes the intended domain and codomain clear. But such context has not been established in this exercise (or list of exercises), and this course is one in which it is imperative that students understand the importance of specifying all three "parts" of a function. We are not in Calc 1 (or 2 or 3) any more, and even though the focus this semester will be on real-valued functions of a real variable, one of the goals of this course is to expand your mind beyond knee-jerk Calc 1 assumptions. Another is to break some bad habits. These things can't wait until the beginning of the second semester to start happening; the learning-curve in the spring would be too steep.
In this Abbott exercise, we can read (part of) the author's mind by reading the next sentence, where it becomes obvious that the domain he has in mind is \(\bf R\) or a subset of \(\bf R\). (Still we do not know what codomain he has in mind. All of \(\bf R\)? The interval \([0,\infty) \)? There's no way of telling.) For the questions asked, the codomain turns out not to be important, but that's really no excuse for ambiguity in the definition of \(f\). And, just as importantly, the reader should never have to read the writer's mind. Having to read the writer's mind means the writing is not good.
There are many domains for which the formula "\(f(x)=x^2\)" makes sense. The sets \( {\bf N, Z, Q, R}, \ \mbox{and}\ {\bf C}\), or any subset of these, are just a few. A few others are the set of \(n\times n\ \mbox{matrices} \) for any \(n\in {\bf N}\), or any group. (If you don't know what a group is, don't worry; you're not expected to in this class. You'd learn about groups in MAS4301 [Abstract Algebra], which is not a prerequisite for this class. But if you're interested, you have my permission to look up what a group is.)- 1.2/9:
- Before starting this exercise, do the reading in part C below. As you'll see there, and as defined in this exercise, the notation "\(f^{-1}(\mbox{some set})\)" means inverse image (= pre-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 whenever you see "\(f^{-1}\)" as part of some notation, you'll entirely miss the point of exercises and exam-problems about inverse images.
- In part (b), take the domain and codomain of \(g\) to be arbitrary sets, not \(\bf R\) (and not necessarily the same set).
- 1.2/11: In 11a, you are not required to prove your "intuitive guess".
- B: Click for non-book problems. Of these, hand in only B1, B2cd, B3b.
- C: Read the handout Difference Between Inverse Functions and Inverse Images (also linked to the Miscellaneous Handouts page, for future reference).
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