Section 1.4 solutions

Section 1.4 solutions #1-10: Find the following sets. U = {a,b,c,d,e} A = {c,d,e} B = {a,c,d} 1) A’ A’ is every element in set U, that is not in set A. Answer: {a,b} 3) 𝐴 ∪ 𝐵 Put sets A and B together in one large set {c,d,e,a,c,d} Remove duplicates and write in a nice order Answer: {a,c,d,e} 5) 𝐴 ∩ 𝐵 This is what A and B have in common. Answer: {c,d} 7) 𝐴′ ∩ 𝐵 A’ = {a,b} B = {a,c,d} I need to find what these two sets have in common. Answer: {a} 9) 𝐴′ ∪ 𝐵 A’ = {a,b} B = {a,c,d} I need to put all of these elements together in one large set. {a,b,a,c,d} Remove duplicates and write in a nice order Answer: {a,b,c,d} #11-20: Find the following sets. U = {1,2,3,4,5} A = {1,2,3} B = {5} 11) A’ A’ is every element in set U that is not in set A. Answer: {4,5} 13) 𝐴 ∪ 𝐵 Put the sets together in one large set {1,2,3,5} I would remove duplicates if there were any. There are no duplicates. This is a fine answer. Answer: {1,2,3,5} 15) 𝐴 ∩ 𝐵 This wants me to find what sets A and B have in common. There is no element in common between the two sets. This answer is the empty set. Answer: ∅ 17) 𝐴′ ∩ 𝐵 A’ = {4,5} B = {5} I need to find what these sets have in common. Answer: {5} 19) 𝐴′ ∪ 𝐵 A’ = {4,5} B = {5} I will put the sets together into a larger set. {4,5,5} Next remove the duplicates and write in a nice order. Answer: {4,5} #21-32: Find the following sets. U = {1,2,3,4,5,6} A = {1,2,3} B = {2,3,4} C = {1,5} 21) 𝐴 ∩ 𝐶 This is what A and C have in common. Answer: {1} 23) 𝐴 ∪ 𝐶 Place the elements of the set together in one large set. {1,2,3,1,5} Remove the duplicates and write in a nice order. Answer: {1,2,3,5} 25) 𝐴 ∩ 𝐵 ∪ 𝐶 First do 𝐴 ∩ 𝐵, this is what A and B have in common. 𝐴 ∩ 𝐵 = {2,3} Now put 𝐴 ∩ 𝐵 and set C together in one large set to find 𝐴 ∩ 𝐵 ∪ 𝐶 {2,3,1,5} There are no duplicates to remove, but I can write this in a nicer order. Answer: {1,2,3,5} 27) 𝐵 ∪ 𝐶 ∩ 𝐴 First find 𝐵 ∪ 𝐶 by placing B and C together in one large set. 𝐵 ∪ 𝐶 = {2,3,4,1,5} There are no duplicates to remove. I can write these in a better order. 𝐵 ∪ 𝐶 = {1,2,3,4,5} Now I can find 𝐵 ∪ 𝐶 ∩ 𝐴 by finding what 𝐵 ∪ 𝐶 and A have in common. 𝐵 ∪ 𝐶 = {1,2,3,4,5} A = {1,2,3} Answer: {1,2,3} 29) 𝐴′ ∩ 𝐵 A’ = {4,5,6} B = {2,3,4} I need to find what these sets have in common. Answer: {4} 31) 𝐴′ ∪ 𝐵 ∩ 𝐶′ First find 𝐴′ ∪ 𝐵 by putting the sets A’ and B together in one large set. A’ = {4,5,6} B = {2,3,4} 𝐴′ ∪ 𝐵 = {2,3,4,5,6} Now I can find 𝐴′ ∪ 𝐵 ∩ 𝐶′ by finding the intersection of 𝐴′ ∪ 𝐵 and C’ I need to find what 𝐴′ ∪ 𝐵 and C’ have in common. 𝐴′ ∪ 𝐵 = {2,3,4,5,6} C’ = {2,3,4,6} Answer: {2,3,4,6} #33-44: Find the following sets. U = {a,b,c,d} A = {a,b,c} B = {b,c,d} C = {a,d} 33) 𝐴 ∩ 𝐶′ Find what A and C’ have in common. A = {a,b,c} C’ = {b,c} Answer: {b,c} 35) 𝐴′ ∪ 𝐶′ A’ = {d} C’ = {b,c} Put these together in one large set. Answer: {b,c,d} 37) 𝐴′ ∩ 𝐵 ∪ 𝐶′ A’ = {d} B = {b,c,d} 𝐴′ ∩ 𝐵 = {d} Now put 𝐴′ ∩ 𝐵 together with C’ to find 𝐴′ ∩ 𝐵 ∪ 𝐶′ 𝐴′ ∩ 𝐵 = {d} C’ = {b,c} Answer: {b,c,d} 39) 𝐵′ ∪ 𝐶′ ∩ 𝐴 First find 𝐵′ ∪ 𝐶′ B’ = {a} C’ = {b,c} 𝐵′ ∪ 𝐶′ = {a,b,c} Now find 𝐵′ ∪ 𝐶′ ∩ 𝐴 by finding what 𝐵′ ∪ 𝐶′ and A have in common. 𝐵′ ∪ 𝐶′ = {a,b,c} A = {a,b,c} Answer: {a,b,c} 41) 𝐴′ ∩ 𝐵′ A’ = {d} B’ = {a} I need to find what these two sets have in common. There are no common elements. The answer is the empty set. Answer: ∅ 43) 𝐴′ ∪ 𝐵′ ∩ 𝐶′ Answer: ∅ #45 – 56: Find the following sets. U = {1,2,3,4,5,6} S = {2,4,6} T = {1,2,4} V = {4,5,6} 45) 𝑆 ∪ (𝑇 ∩ 𝑉) First work on the inside of the parenthesis. (𝑇 ∩ 𝑉) = {4} Now put (𝑇 ∩ 𝑉) and S together into one large set to find 𝑆 ∪ (𝑇 ∩ 𝑉) S = {2,4,6} (𝑇 ∩ 𝑉) = {4} Answer: {2,4,6} 47) (𝑆 ∪ 𝑇)′ First find 𝑆 ∪ 𝑇 = {1,2,4,6} Now find the complement of this to get my answer. (𝑆 ∪ 𝑇)′ = {3,5} Answer: {3,5} 49) 𝑆 ∩ (𝑉 ∩ 𝑇 ′ ) First find (𝑉 ∩ 𝑇 ′ ) V = {4,5,6} T’ = {3,5,6} (𝑉 ∩ 𝑇 ′ ) = {5,6} Now intersect this with S to find 𝑆 ∩ (𝑉 ∩ 𝑇 ′ ) S = {2,4,6} (𝑉 ∩ 𝑇 ′ ) = {5,6} 𝑆 ∩ (𝑉 ∩ 𝑇 ′ ) = {6} Answer: {6} 51) (𝑆 ′ ∪ 𝑉 ′ ) ∩ 𝑇 First find (𝑆 ′ ∪ 𝑉 ′ ) S’ = {1,3,5} V’ = {1,2,3} (𝑆 ′ ∪ 𝑉 ′ ) = {1,2,3,5} Now find (𝑆 ′ ∪ 𝑉 ′ ) ∩ 𝑇 by finding what (𝑆 ′ ∪ 𝑉 ′ ) have in common. (𝑆 ′ ∪ 𝑉 ′ ) = {1,2,3,5} T = {1,2,4} Answer: {1,2} 53) 𝑇 ∪ 𝑉′ ∪ 𝑆′ First find 𝑇 ∪ 𝑉′ T = {1,2,4} V’ = {1,2,3} 𝑇 ∪ 𝑉′ = {1,2,3,4} Now find 𝑇 ∪ 𝑉′ ∪ 𝑆′ by putting 𝑇 ∪ 𝑉′ and S’ into a larger set. 𝑇 ∪ 𝑉′ = {1,2,3,4} S’ = {1,3,5} Answer: {1,2,3,4,5} 55) (𝑉 ∩ 𝑇)′ ∪ 𝑆 First find (𝑉 ∩ 𝑇)= {4} Now we can find (𝑉 ∩ 𝑇)′= {1,2,3,5,6} Now put (𝑉 ∩ 𝑇)′ and S together to get(𝑉 ∩ 𝑇)′ ∪ 𝑆 (𝑉 ∩ 𝑇)′= {1,2,3,5,6} S = {2,4,6} Answer: {1,2,3,4,5,6}