class lecture notes 8

136 Chapter 7. Logic, Sets, and Counting Techniques (LECTURE NOTES 8)(f) True / False (E 1 E 2 ) ′ = E ′ 1 ∪E ′ 2 (DeMorgan’s Laws)3. Venn diagrams and counting.Let A = {a,b,c,d,e,f},B = {b,d,e,h},C = {a,e}and the universal set is U = {a,b,c,d,e,f,g,h,i,j}. Let n be “number”.AcbdfeaUCBhgi jFigure 7.8(Venn diagrams and counting)(a) n(A) = n{a,b,c,d,e,f} = 5 / 6(b) n(B) = n{b,d,e,h} = 1 / 2 / 3 / 4,(c) n(C) = n{a,e} = 1 / 2 / 3 / 4,(d) n(A∩B) = n{b,d,e} = 1 / 2 / 3 / 4,(e) n(A∩C) = n{a,e} = 1 / 2 / 3 / 4,(f) n(B ∩C) = n{e} = 1 / 2 / 3 / 4,(g) n(A∩B ∩C) = n{e} = 1 / 2 / 3 / 4,(h) n(A∪B) = n(A)+n(B)−n(A∩B) == n{a,b,c,d,e,f}+n{b,d,e,h}−n{b,d,e} = 5 / 6 / 7 / 8,(i) n(A∪C) = n{a,b,c,d,e,f} = 5 / 6 / 7 / 8,(j) n(B ∪C) = n{a,b,d,e,h} = 5 / 6 / 7 / 8,(k) n(A∪B ∪C) == n(A)+n(B)+n(C)−n(A∩B)−n(A∩C)−n(B∩C)+n(A∩B∩C) == n{a,b,d,c,e,f,h} = 5 / 6 / 7 / 8 / 9,(l) n(A ′ ) = n(U)−n(A) = n{g,h,i,j} = 1 / 2 / 3 / 4,(m) n(B ′ ) = n{a,c,f,g,i,j} = 5 / 6 / 7 / 8,(n) n(C ′ ) = n{b,c,d,f,g,h,i,j} = 5 / 6 / 7 / 8,(o) n(A ′ ∩B ′ ) = n{g,i,j} = 1 / 2 / 3 / 4,