Please help with the boxes marked with an xchoices areAssociative law for *Commutative law for *Complement law for *Distributive law of * over +Identity law of +Universal bound law for * 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) · (ā · b) = 0.(a + b). (ā.b)(a. b) (a + b)====(a.b).a +((ā. 5). b)((-a) a) + ((a.). b)b)(b. (ā-a)) + (ā- (5.b))·b. (a.a)) +(ā· (b.5))(b 0) + (a. 0)= 0 + 0= 0by the commutative law forby the distributive law of over +by the commutative law for.by the distributive law of over +by the identity law of +by the associative law forby the commutative law forby the complement law for+X+ XX

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Part 1: Proof that (a + b) · (à · b) = 0. (a + b). (a. b) = = = = = = (a. b)(a + b) ((ā· b). a) + ((a. b) (5. (a.a) (b ·ā)·a) + ((ā· b) b. (a.a)) +(ā. (5.b)) ·ā)) + (ā· (b · ·5)) (5.0) + (ā.0) b) = 0 + 0 = 0 b b by the commutative law for by the distributive law of over + → by the commutative law for by the distributive law of over + X by the identity law of + by the associative law for by the commutative law for by the complement law for + X X +x +X

Part 1: Proof that (a + b) · (à · b) = 0. (a + b). (a. b) = = = = = = (a. b)(a + b) ((ā· b). a) + ((a. b) (5. (a.a) (b ·ā)·a) + ((ā· b) b. (a.a)) +(ā. (5.b)) ·ā)) + (ā· (b · ·5)) (5.0) + (ā.0) b) = 0 + 0 = 0 b b by the commutative law for by the distributive law of over + → by the commutative law for by the distributive law of over + X by the identity law of + by the associative law for by the commutative law for by the complement law for + X X +x +X

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 6TFE: True or False Label each of the following statements as either true or false. 6. Let . The empty set...

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