134 Chapter 7. Logic, Sets, and Counting Techniques (LECTURE NOTES 8)U U U1 E E E4 115442 355G 2 32 3GG6F 766F 7F 7888(a) (b) (c)Figure 7.6(Venn diagrams)(a) Shaded region figure (a). Fill in blank.Set region labelsE 1,2,3,4F 2,3,6,7G 3,4,5,6(E ∩F)∪(E ∩G)(b) Shaded region figure (b). Fill in blank.Set region labelsE 1,2,3,4F 2,3,6,7G 3,4,5,6(E −G)∪(F −G)∪(E ∩F)(c) Shaded region figure (c). Fill in blank.Set region labelsE 1,2,3,4F 2,3,6,7G 3,4,5,6(E ∪F) ′2. More Venn diagrams and labeling. LetE 1 = {a,b,c,d,e,f},E 2 = {e,f,g,h},E 3 = {i}and the universal set is U = {a,b,c,d,e,f,g,h,i,j}.
Section 4. Application of Venn Diagrams (LECTURE NOTES 8) 135E1a b c de fiEU3E2g hjFigure 7.7(Venn diagrams and counting)(a) Since E 1 ∩E 2 = E 1 E 2 ={e,f} / {a,b,c,d} / {a,b,c,d,i,j}then (E 1 E 2 )E 3 = {e,f} / {a,b,c,d} / ØOn the other hand, sinceE 2 E 3 = {e,f} / {a,b,c,d} / Øthen E 1 (E 2 E 3 ) = {e,f} / {a,b,c,d} / ØSo (E 1 E 2 )E 3 = E 1 (E 2 E 3 ) (associative law)(b) True / False E 1 ∪(E 2 ∪E 3 ) = (E 1 ∪E 2 )∪E 3 (associative law)(c) Since E 1 ∪E 2 ={e,f} / {a,b,c,d} / {a,b,c,d,e,f,g,h}then (E 1 ∪E 2 )E 3 = {e,f} / {a,b,c,d} / Øon the other hand, sinceE 1 E 3 = {e,f} / {a,b,c,d} / Øand E 2 E 3 = {e,f} / {a,b,c,d} / Øthen E 1 E 3 ∪E 2 E 3 = {e,f} / {a,b,c,d} / Øand so (E 1 ∪E 2 )E 3 = E 1 E 3 ∪E 2 E 3 (distributive law)(d) True / False(E 1 ∪E 3 )(E 2 ∪E 3 ) = E 1 E 2 ∪E 1 E 3 ∪E 3 E 2 ∪E 3 E 3= E 1 E 2 ∪Ø∪Ø∪E 3= E 1 E 2 ∪E 3(e) Since E 1 ∪E 2 ={e,f} / {a,b,c,d} / {a,b,c,d,e,f,g,h}then (E 1 ∪E 2 ) ′ = {e,f} / {a,b,c,d} / {i,j}on the other hand, sinceE 1 ′ = {g,h,i,j} / {a,b,c,d} / Øand E 2 ′ = {e,f} / {a,b,c,d} / {a,b,c,d,i,j}then E 1E ′ 2 ′ = {e,f} / {a,b,c,d} / {i,j}and so (E 1 ∪E 2 ) ′ = E 1 ′ E′ 2 (DeMorgan’s Laws).