(a) Prove the following statement using the element method for proving that a set equals the empty set: For all sets A and B, AN(B – A) = (). (b) Use the properties in Theorem 6.2.2 shown below to prove the statement in part (a), Be sure to give a reason for every step.

Elements Of Modern Algebra
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Chapter7: Real And Complex Numbers
Section7.1: The Field Of Real Numbers
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(a) Prove the following statement using the element method for proving that a
set equals the empty set: For all sets A and B, AN(B – A) = ().
(b) Use the properties in Theorem 6.2.2 shown below to prove the statement in
part (a). Be sure to give a reason for every step.
Transcribed Image Text:(a) Prove the following statement using the element method for proving that a set equals the empty set: For all sets A and B, AN(B – A) = (). (b) Use the properties in Theorem 6.2.2 shown below to prove the statement in part (a). Be sure to give a reason for every step.
Theorem 6.2.2 Set Identities
Let all sets referred to below be subsets of a universal set U.
1. Commutative Laws: For all sets A and B,
(a) AUB = BUA and (b) ANB = BNA.
2. Associative Laws: For all sets A, B, and C,
(a) (A U B) U C = AU (BUC) and
(b) (A N B) NC = AN (BN C).
3. Distributive Laws: For all sets A, B, and C,
(a) AU (BN C) = (A U B) N (A UC) and
(b) AN (BUC) = (A N B) U (AN C).
4. Identity Laws: For every set A,
(a) AUØ = A and (b) A N U = A.
5. Complement Laws: For every set A,
(a) AUA° = U and (b) A NA° = Ø.
6. Double Complement Law: For every set A,
(A) = A.
7. Idempotent Laws: For every set A,
(a) AUA = A and (b) A NA = A.
8. Universal Bound Laws: For every set A,
(a) AUU= U and (b) A nØ=Ø.
9. De Morgan's Laws: For all sets A and B,
(a) (AU B) = A“N B° and (b) (AN B)° = A° UB".
10. Absorption Laws: For all sets A and B,
(a) AU (A N B) = A and (b) AN (AU B) = A.
11. Complements of U and Ø:
(a) UC = Ø and (b) Ø = U.
12. Set Difference Law: For all sets A and B,
A -B = ANBº.
Transcribed Image Text:Theorem 6.2.2 Set Identities Let all sets referred to below be subsets of a universal set U. 1. Commutative Laws: For all sets A and B, (a) AUB = BUA and (b) ANB = BNA. 2. Associative Laws: For all sets A, B, and C, (a) (A U B) U C = AU (BUC) and (b) (A N B) NC = AN (BN C). 3. Distributive Laws: For all sets A, B, and C, (a) AU (BN C) = (A U B) N (A UC) and (b) AN (BUC) = (A N B) U (AN C). 4. Identity Laws: For every set A, (a) AUØ = A and (b) A N U = A. 5. Complement Laws: For every set A, (a) AUA° = U and (b) A NA° = Ø. 6. Double Complement Law: For every set A, (A) = A. 7. Idempotent Laws: For every set A, (a) AUA = A and (b) A NA = A. 8. Universal Bound Laws: For every set A, (a) AUU= U and (b) A nØ=Ø. 9. De Morgan's Laws: For all sets A and B, (a) (AU B) = A“N B° and (b) (AN B)° = A° UB". 10. Absorption Laws: For all sets A and B, (a) AU (A N B) = A and (b) AN (AU B) = A. 11. Complements of U and Ø: (a) UC = Ø and (b) Ø = U. 12. Set Difference Law: For all sets A and B, A -B = ANBº.
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