class lecture notes 8

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).