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Assume that B is a Boolean algebra with operations + and. Prove the following statement. De Morgan's law for +: For all a and b in B, a +b=a.b. Proof: Suppose B is a Boolean algebra and a and b are any elements of B. [We must show that a + b = a.b.] Part 1: Proof that (a + b) (a - b) = 0. (a + b) (ab) = (3.b)(a + b) ((a. b) a) + ((a.5) b) = ((b-ā). a) + ((ā-5). b) = = (b (aa))+((6.b)) = (b. (aa)) + (a (b.5)) = (5.0) + (ā.0) = 0+0 = 0 Part 2: Proof that (a + b) + (ab) = 1. Write a proof for Part 2 as a free response. Be s Choose File no file selected This answer has not been graded yet. --Select-- ---Select-- ---Select--- ---Select-- ---Select-- ---Select--- --Select--- ✓-Select--- by the associative law for by the commutative law for by the complement law for by the distributive law of over + by the identity law of + by the universal bound law for # # # # # + ep. (Submit a file with a maximum size of 1 MB.) Then return to finish Part 3. Part 3: Conclusion. Because both (a + b)(a 5) = 0 and (a + b) + (-5) = 1, it follows from the ---Select--- . that a+b=a.5.

Assume that B is a Boolean algebra with operations + and. Prove the following statement. De Morgan's law for +: For all a and b in B, a +b=a.b. Proof: Suppose B is a Boolean algebra and a and b are any elements of B. [We must show that a + b = a.b.] Part 1: Proof that (a + b) (a - b) = 0. (a + b) (ab) = (3.b)(a + b) ((a. b) a) + ((a.5) b) = ((b-ā). a) + ((ā-5). b) = = (b (aa))+((6.b)) = (b. (aa)) + (a (b.5)) = (5.0) + (ā.0) = 0+0 = 0 Part 2: Proof that (a + b) + (ab) = 1. Write a proof for Part 2 as a free response. Be s Choose File no file selected This answer has not been graded yet. --Select-- ---Select-- ---Select--- ---Select-- ---Select-- ---Select--- --Select--- ✓-Select--- by the associative law for by the commutative law for by the complement law for by the distributive law of over + by the identity law of + by the universal bound law for # # # # # + ep. (Submit a file with a maximum size of 1 MB.) Then return to finish Part 3. Part 3: Conclusion. Because both (a + b)(a 5) = 0 and (a + b) + (-5) = 1, it follows from the ---Select--- . that a+b=a.5.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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universal bound laws for + and -

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uniqueness of the complement law

Assume that B is a Boolean algebra with operations + and. Prove the following statement. De Morgan's law for +: For all a and b in B, a + b = a·b. .6.1 Proof: Suppose B is a Boolean algebra and a and b are any elements of B. [We must show that a + b = Part 1: Proof that (a + b) · (ā.b) = 0. (a + b). (ā.5) (ā.b). (a + b) = = (ab) a) + ((a.b). b • = ((6.a) a) + ((ab). b) = (b · (aa)) + (ā· (b. b)) = (b · (a ·ā)) + (ā· (b.5)) (5.0) + (ā.0) = 0 + 0 = 0 Part 2: Proof that (a + b) + (à · b) = 1. Write a proof for Part 2 as a free response. Be s Choose File no file selected This answer has not been graded yet. ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- by the associative law for. by the commutative law for. by the complement law for by the distributive law of over + by the identity law of + by the universal bound law for + + + + ep. (Submit a file with a maximum size of 1 MB.) Then return to finish Part 3. Part 3: Conclusion. Because both (a + b) · (à · 5) = 0 and (a + b) + (ā. b) = 1, it follows from the ---Select--- that a + b = a.b.
Transcribed Image Text:Assume that B is a Boolean algebra with operations + and. Prove the following statement. De Morgan's law for +: For all a and b in B, a + b = a·b. .6.1 Proof: Suppose B is a Boolean algebra and a and b are any elements of B. [We must show that a + b = Part 1: Proof that (a + b) · (ā.b) = 0. (a + b). (ā.5) (ā.b). (a + b) = = (ab) a) + ((a.b). b • = ((6.a) a) + ((ab). b) = (b · (aa)) + (ā· (b. b)) = (b · (a ·ā)) + (ā· (b.5)) (5.0) + (ā.0) = 0 + 0 = 0 Part 2: Proof that (a + b) + (à · b) = 1. Write a proof for Part 2 as a free response. Be s Choose File no file selected This answer has not been graded yet. ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- ---Select--- by the associative law for. by the commutative law for. by the complement law for by the distributive law of over + by the identity law of + by the universal bound law for + + + + ep. (Submit a file with a maximum size of 1 MB.) Then return to finish Part 3. Part 3: Conclusion. Because both (a + b) · (à · 5) = 0 and (a + b) + (ā. b) = 1, it follows from the ---Select--- that a + b = a.b.
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