Last updated Sat Oct 3 19:20 EDT 2020
Due-date: Wednesday, 10/7/20
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 exercises:
- Section 2.1/ 1–3, 10b , 11 (we already did the first part of 11a in class), 12–14, 16abc, 20, 22, 23, 26
- Section 2.2/ 1, 10, 18
- Section 2.3/ 5, 6, 10, 14
- Section 2.4/ 1, 2, 4, 8
- Section 2.5/ 9
- Section 3.1/ 5ad, 8 (just the "Give an example" part; the first part of the problem was done in the lecture notes), 12 (read Example , 3.1.6d first)
- Section 3.2/ 1, 2, 5b, 6, 12, 13
- Section 3.3/ 1, 4, 6, 7
- Section 3.4/ 4, 12 (#5 has been replaced by non-book problem B1)
Of the B& S problems, hand in only these: 2.3/5c (formal proof not required, but state your reasoning); 3.1/5d; 3.2/1b, 2, 12 (you're not required to use the same sequences in 2b as in 2a); 3.4/4a
Notes on some of these problems:
- In 2.1/ 1–2, the point is to do these by using only field properties of \({\bf R}\) ("2.1.1 Algebraic properties of \({\bf R}\)", and the consequences of these properties that we already proved from these in class and/or are proven in Theorems 2.1.2 and 2.1.3). In this problem, anything like "This fact is true because I was taught it in high school (or middle school)" is not a valid argument—essentially, it's circular reasoning. What you're doing here is showing that various "rules of algebra" are consequences of the field properties. Make sure you don't fall victim to any of the pitfalls mentioned in the What is a proof handout, pp. 3–6, especially " 'Proof' by lack of contradiction", " 'Proof' by notation", and " 'Proof' by lack of imagination".
- Similarly, in 2.1/ 3, the point is to do these according to the instructions: "justifying each step by referring to an appropriate property [listed among the field properties] or theorem." You all know what steps to take, and could solve any of these equations in five seconds or less, but that's not the point of this exercise.
- Similarly, 2.1/ 16abc should be done by using only the field and order properties of \( {\bf R}\) we listed, and/or consequences that we derived. Drawing a diagram of the real line, with plus or minus signs over/under various intervals, does not constitute such a method (though you may certainly use it to help you find a valid method).
- For 2.1/ 11a, we already did the first part in class; you need only do the second part (proving \( 1/(1/a) =a\)). You should find this proof very similar to your proof of 2.1/ 1b.
- In 2.1/ 26 there's a typo: the equality on the second line should be "\( (a^m)^n = a^{mn}\)."
- B: Click for non-book problems. Of these, hand in only B1, B2.
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