MAA 4211, Section 1216 (26002) — Fall 2021
Assignment 2

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Last updated Mon Sep 27 14:19 EDT 2021

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Due-date: Friday, 10/1/21

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.

• A: Abbott exercises. See notes and additional instructions below this list before starting these problems.
  • 1.3/ 1–4, 6, 8, 11
  • 1.4/ 4
  • 1.5/ 1, 3b
  Of the Abbott problems, hand in only these:
  • 1.3/ 3a, 4, 8d, 11bc
  • 1.5/ 1

    Notes and additional instructions on some of these problems:

    • At the end of Section 1.3, between the heading "Exercises" and the statement of Exercise 1.3.1, the author should have said: "In the following exercises, 'set' always means 'subset of \({\bf R}\)'."

    • 1.3.4(a).
      • Insert the words "in terms of \(\sup(A_1)\) and \(\sup(A_2)\)" after "\(\sup(A_1\cup A_2)\)".

      • Prove your answers.

    • 1.3.4(b). If your answer is yes, prove it. If your answer is no, provide a counterexample.

    • 1.3.6(d). The "other proof" you're constructing is allowed to share some steps with the first proof that the earlier parts of this problem lead you to. For example, it's fine if one of your steps shows the fact given in 1.3.6(a).

    • 1.3.8. Change "without proofs" to "with proofs". (I am not likely to ask you to hand in all of these, but for any that I do ask you to hand in, I'll require proof(s).)

    • 1.3.11.
      • (a) (i) Replace "bounded" by "bounded above". (A set \(C\subseteq {\bf R}\) is called bounded if \(C\) is bounded above and bounded below.)

             (ii) If \(A\) and \(B\) are nonempty, bounded below, and \(A\subseteq B\), what order-relation between \(\inf(A)\) and \(\inf(B)\) is automatically satisfied?

             (iii) If \(C\) is bounded and non-empty, what order-relation between \(\inf(C)\) and \(\sup(C)\) is automatically satisfied?

             (iv) Assuming that \(A\) and \(B\) are nonempty and bounded, and \(A\subseteq B\), combine the results of (i), (ii), and (iii) to fill in the blanks appropriately in this string of inequalities:
            (blank)\( \leq \) (blank) \( \leq\) (blank) \( \leq\) (blank).

      • (b): Better wording would have been "If \(A\) and \(B\) are sets for which \( \sup(A)\) and \(\inf(B)\) exist and satisfy \( \sup(A)<\inf(B)\), then ..."

      • (c) Assume \(A\) and \(B\) are nonempty.
            Note: When a conclusion states an equality or inequality between numbers that, in general, may or may not exist (and have not been assumed to exist), then the conclusion is interpreted as saying: both sides of the equation/inequality exist, and satisfy the indicated relation. This applies to the inequality "\(\sup(A)<\inf(B)\)" in this problem. (Thus, you should interpret the portion of the sentence after the comma as: "then \(\sup(A)\) and \(\inf(B)\) exist, and satisfy \(\sup(A)<\inf(B)\)."

    • 1.5.1. (i) Remember that Abbott's "countable" is my "countably infinite." (ii) When combined with the fact that every subset of a finite set is finite (non-book problem B4), Theorem 1.5.7 is equivalent to the following statement: Every subset of an at-most countable set is at most countable. (Remember that "at most countable" means "finite or countably infinite", for which Abbott would say "finite or countable".)

    • 1.5.1, continued. For what you are handing in, treat the problem as if it had been worded as follows: Let \(B\) be a countably infinite set and assume that \(A\subseteq B\) is an infinite set. Construct, with proof, a bijection \(g:{\bf N}\to A\). (I.e. follow Abbott's sketch for constructing a function \(g:{\bf N}\to A\) a certain way, and show that this construction does, indeed, yield a bijection from \({\bf N}\) to \(A\).)

  • B: Click for non-book problems. Of these, hand in only B1, B5, B7(a). For B5, you may assume the results of B1–B4, in addition to what the first note at the end of B5 says you may assume. See also the "But sometimes ..." part of the third note at the end of B5.

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