Due Wednesday, April 16
- Sect. 8:
- Do Exercise 8.4(b)** completely (even if you have done it before) AFTER solving the following problems*.
- For each real number a, show that the open intervals (0, ∞) and (a, ∞) are equinumerous.
- For each real number a, show that the open intervals (-∞, -a) and (a, ∞) are equinumerous.
* These problems require you to define a function which is a bijection, and you should also PROVE that it is a bijection. Be sure that you know how (i) to define a function between two given sets and (ii) to prove that it is a bijection.
See Defining Functions and Function Warnings.
** READ THE INSTRUCTIONS IN THE BOOK for part (b). Explain why you can “use part (a)” in Exercise 8.4 only if the intervals in (b) are bounded. (Roughly speaking, a bounded interval is an interval which does not have an infinity as an endpoint.) Give a complete proof of (b) by using previous results, including part (a), and the problems above, and relevant parts of Exercise 8.3. (You are specifically allowed to use Ex. 8.3, even though it has not been assigned as homework. But since you are asked to give a complete proof of part (b), do NOT use your previous solution of 8.4(b).)
USING PREVIOUS RESULTS does not mean redoing previous results, or noting that this problem has a proof which is similar to a previous problem. It means using the RESULTS of the previous problems, regardless of whether you can prove them.
NOTE: Be sure READ THE INSTRUCTIONS ABOVE and to read Homework Format and Homework Writing Policy BEFORE writing up your solutions to be turned in.
NOTE: As stated on the Course Home Page, all due dates are tentative. Assignments, or parts of assignments, may be postponed to a later date.
Last modified Apr 12, 2008 11:06 PM
Go to Lesson 44 (due Apr 11).
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