5. Are the following true or false, and prove your answers:(a) If A is compact and B is bounded, then An B is compact.(b) If A is compact and B is closed, then An B is compact.(c) If A is compact and B is bounded, then AUB is compact.(d) If A is compact and B is closed, then AUB is compact.

Answered: Image /qna-images/answer/41610467-5e37-4910-9162-dd7b8b3195b2.jpg

Answered: 5. Are the following true or false, and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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a) Give an example of an infinitely large set A and a functiond: A × A → R≥0 so that d satisfies the triangle inequality, butis degenerate (i.e. there exist several x, y ∈ A with d(x, y) = 0).(b) Give an example of an infinitely large set A and a functiond: A × A → R≥0 so that d satisfies the triangle inequality, butis not symmetrical.(c) Give an example of an infinitely large set A and afunction d: A×A → R≥0 so that d does satisfy the triangle inequality,but at the same time degenerate and not symmetrical. I always struggle with questions like this where I have to find a set, please if able provide some insight on how to approach this best. Thank you very much.
1. Let DC R. Write down precise mathematical formulations of the following statements, using quantifiers (V, 3): (a) D is bounded above. (b) D has a maximum element. (c) D does not have a maximum element.
2. Find the infimum, the supremum, the minimum and the maximum (if they exist) of the following sets: (i) A = {1,3,5,7.9,11} (-1)" (iii) A 4. 2 3 4 (v) A={x eR:-4<2} 3. Which of the sets in (2) are bounded?
I don't understand this question at all. I've asked it a few times, and have gotten what look like a correct solutions, but I don't understand the reasoning behind many of the steps. I'm asking it again, hoping to get an even more thorough explanation. I'm sorry, it just isn't making sense to me. Thanks for your time.
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5. Are the following true or false, and prove your answers: (a) If A is compact and B is bounded, then An B is compact. (b) If A is compact and B is closed, then An B is compact. (c) If A is compact and B is bounded, then AUB is compact. (d) If A is compact and B is closed, then AUB is compact.