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Does each element of G have a unique inverse under the operation U? If so, which part of Proposition 7.2.3 enabled you to draw this conclusion? If not, provide a counterexample.

Does each element of G have a unique inverse under the operation U? If so, which part of Proposition 7.2.3 enabled you to draw this conclusion? If not, provide a counterexample.

Advanced Engineering Mathematics
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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Proposition 7.2.3. Let A, B, and C be subsets of a universal set U. Then 1. AUA' = U and AnA' = 0 2. AUA = A, An A = A, and A \ A = 0; 3. AUØ = A and An0 = 0; 4. AUU = U and ANU = A; 5. AU(BUC) = (AUB)UC and An (BnC) = (An B)nC; 6. AUB = BU A and An B = BN ; 7. AU(BNC) = (AUB)N(AUC) and (BNC)UA= (BUA)N(CUA); 8. An(BUC) = (ANB)U(ANC) and (BUC)NA= (BNA) U (CN A).
Transcribed Image Text:Proposition 7.2.3. Let A, B, and C be subsets of a universal set U. Then 1. AUA' = U and AnA' = 0 2. AUA = A, An A = A, and A \ A = 0; 3. AUØ = A and An0 = 0; 4. AUU = U and ANU = A; 5. AU(BUC) = (AUB)UC and An (BnC) = (An B)nC; 6. AUB = BU A and An B = BN ; 7. AU(BNC) = (AUB)N(AUC) and (BNC)UA= (BUA)N(CUA); 8. An(BUC) = (ANB)U(ANC) and (BUC)NA= (BNA) U (CN A).
Exercise 7.3.2. Let G be the set of subsets of the set {a, b, c}. (a) Does the set G with the operation U have the closure property? Justify your answer. (b) Does the set G with the operation U have an identity? If so, what is it? Which part of Proposition 7.2.3 enabled you to draw this conclusion? (c) Is the operation U defined on the set G associative? Which part of Proposition 7.2.3 enabled you to draw this conclusion? (d) Is the operation U defined on the set G commutative? Which part of Proposition 7.2.3 enabled you to draw this conclusion? (e) Does each element of G have a unique inverse under the operation U? If so, which part of Proposition 7.2.3 enabled you to draw this conclusion? If not, provide a counterexample. (f) Is the set G a group under the U operation? Justify your answer.
Transcribed Image Text:Exercise 7.3.2. Let G be the set of subsets of the set {a, b, c}. (a) Does the set G with the operation U have the closure property? Justify your answer. (b) Does the set G with the operation U have an identity? If so, what is it? Which part of Proposition 7.2.3 enabled you to draw this conclusion? (c) Is the operation U defined on the set G associative? Which part of Proposition 7.2.3 enabled you to draw this conclusion? (d) Is the operation U defined on the set G commutative? Which part of Proposition 7.2.3 enabled you to draw this conclusion? (e) Does each element of G have a unique inverse under the operation U? If so, which part of Proposition 7.2.3 enabled you to draw this conclusion? If not, provide a counterexample. (f) Is the set G a group under the U operation? Justify your answer.
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