Consider the setsA = {5n+3 | n ≤N},B = {3n+2 | n ≤ N}.(a) List three elements in A.(b) List three elements in B.(c) Is it true that AC B? And is it true that BCA? Can you explain why?(d) How many elements do you think there are in AUB? Can you explain why you think so?(e) How many elements do you think there are in An B? Can you explain why you think so?

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Answered: Consider the sets A = {5n+3 | n ≤N}, B…

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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Another Field We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from 13 2 the set Q). For example, +¹3√2, 7.2-6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √√3 nor are rational numbers). A number such 3+5√2 would be in the set Q(√2) since 3+√2 can be re-written in the form 7 a+b√2 using the following steps: 7 7 3-5√2 21-35√2 21-35√2 21 - 35√2 21 35 == + √2 3+5√2 3+5√2 3-5√2 9 - 25(2) -41 41 41 = * Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and * is the ordinary multiplication operation. So, you need to show that: a. (Q(√2), +, *) is a commutative ring, and b. (Q(√2) - {0}, *) is a group.

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