Chapter 5 Section 3 Venn Diagram and Counting

Exercise 13 (page 222) Given: n(U) = 20 n(S) = 12 n(T) = 14 n(S ∩ T ) = 18 Problem, none of the values above corresponds to any basic of the regions

Exercise 13 Solution Use the inclusion-exclusion principle n( S U T ) = n( S ) + n( T ) – n( S ∩ T ) 18 = – n( S ∩ T ) n( S ∩ T ) = 8

Exercise 13 Venn Diagram Basic region IV = 20 – ( ) = 2 S 2 4 U 8 T 6

Exercise 16 (page 222) Given: n(S) = 9 n(T) = 11 n(S ∩ T ) = 5 ** n(S´ ) = 13 We can fill in all the basic regions except for basic region I.

Exercise 16 Solution Use the following fact: n( S ) + n( S ´ ) = n( U ) = n( U ) n( U ) = 22 Recall that all 4 basic regions must add up to n( U ). Thus: Basic region IV = 22 – ( ) = 7 With this information we then can fill in the Venn Diagram.

Exercise 16 Venn Diagram S 7 4 U 5 T 6

Exercise 23 (page 222) First we need to define our sets: Let: S = { Students who like rock music } T = { Students who like hip-hop music } “Survey of 70 …students”n(U) = 70 “35 students like rock music”n(S) = 35 “15 students like hip-hop”n(T) = 15 “5 liked both”n(S ∩ T) = 5 ** Since n(S ∩ T) is basic region I in the two set Venn Diagram, we can start filling in the Venn Diagram. ( ** means that this is a basic region in the Venn Diagram)

Exercise 23 Venn Diagram Basic region IV = 70 – ( ) = 25 S U 5 T 10

Exercise 31 (page 223) First we need to define our sets: Let: U = { Students in Finite Math class } M = { Male students in Finite Math class } B = { Business students in Finite Math class } F = { First year students in Finite Math class }

Exercise 31 Given “35 students in class”n(U) = 35 “22 are male students”n(M) = 22 “19 are business students”n(B) = 19 “27 are first-year students”n(F) = 27 “17 are male first-year”n( M ∩ F ) = 17 “15 are first-year businessn( B ∩ F ) = 15 “14 are male business”n( M ∩ B ) = 14 “11 are male first-year business”n( M ∩ B ∩ F ) = 11 ** ( ** means that this is a basic region in the Venn Diagram)

Exercise 31 Venn Diagram Basic Region VIII = 35 – ( ) = 2 B 2 2 U 3 F 1 M