Problem 6. (ii) please! (ii) Use (i) to show that if |a – b| < e for all e> 0, then a = b.3. Let ACR. Define -A ={-a:a € A}. Suppose that A is non-empty and bounded below. Show thatinf(A) = -sup(-A)4. Let A = {„1 inEN}. Prove that sup(A) = 1, inf(A) = }.%3D5. (i) Let A, BCR be sets which are bounded above, such that A C B. Show that sup(A) < sup(B).(ii) Let A, BCR such that sup(A) < sup(B). Show that there exists be B that is an upper bound of A.Show that this result does not hold if we instead assume that sup(A) < sup(B).6. For A, B CR, defineA + B = {a+b: a € A, b e B}A -B = {a ·b: a € A, b E B}(i) Determine {3, 1,0} + {2,0, 2, 1} and {3, 1,0} {2,0, 2, 1}(ii) Assume that sup(A) and sup(B) exist. Prove that sup(A + B) = sup(A) + sup(B).(iii) Give an example of sets A, B where sup(A· B) + sup(A) - sup(B)Warm-up Problems, Not for credit:1. Let F be any field. Prove that both the additive and multiplicative identities in F are unique.2. Given an ordered field F, we saw that it has a set of positive elements P satisfying certain two conditions.24°C 맑음 ^ qx 8면 여기에 입력하십시오.LG

Answered: Image /qna-images/answer/a6729c46-21af-4524-bbcf-086fbd29075b.jpg

6. For A, B CR, define A+B = {a+b: a E A, b e B} A B = {a·b: a E A, b E B} %3! (i) Determine {3,1,0} + {2,0, 2, 1} and {3, 1,0} · {2,0, 2, 1} (ii) Assume that sup(A) and sup(B) exist. Prove that sup(A+ B) = sup(A)+ sup(B). (iii) Give an example of sets A, B where sup(A · B) + sup(A) - sup(B) %3D

6. For A, B CR, define A+B = {a+b: a E A, b e B} A B = {a·b: a E A, b E B} %3! (i) Determine {3,1,0} + {2,0, 2, 1} and {3, 1,0} · {2,0, 2, 1} (ii) Assume that sup(A) and sup(B) exist. Prove that sup(A+ B) = sup(A)+ sup(B). (iii) Give an example of sets A, B where sup(A · B) + sup(A) - sup(B) %3D

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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Question

Problem 6. (ii) please!

(ii) Use (i) to show that if |a – b| < e for all e> 0, then a = b.
3. Let ACR. Define -A ={-a:a € A}. Suppose that A is non-empty and bounded below. Show that
inf(A) = -sup(-A)
4. Let A = {„1 inEN}. Prove that sup(A) = 1, inf(A) = }.
%3D
5. (i) Let A, BCR be sets which are bounded above, such that A C B. Show that sup(A) < sup(B).
(ii) Let A, BCR such that sup(A) < sup(B). Show that there exists be B that is an upper bound of A.
Show that this result does not hold if we instead assume that sup(A) < sup(B).
6. For A, B CR, define
A + B = {a+b: a € A, b e B}
A -B = {a ·b: a € A, b E B}
(i) Determine {3, 1,0} + {2,0, 2, 1} and {3, 1,0} {2,0, 2, 1}
(ii) Assume that sup(A) and sup(B) exist. Prove that sup(A + B) = sup(A) + sup(B).
(iii) Give an example of sets A, B where sup(A· B) + sup(A) - sup(B)
Warm-up Problems, Not for credit:
1. Let F be any field. Prove that both the additive and multiplicative identities in F are unique.
2. Given an ordered field F, we saw that it has a set of positive elements P satisfying certain two conditions.
24°C 맑음 ^ qx 8
면 여기에 입력하십시오.
LG

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