Chapter 5 Section 3

1. Chapter 5 Section 3 Venn Diagram and Counting

2. 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

3. Exercise 13 Solution • Use the inclusion-exclusion principle n( SUT ) = n( S ) + n( T ) – n( S∩T ) 18 = 12 + 14 – n( S∩T ) n( S∩T ) = 8

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

5. 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.

6. Exercise 16 Solution • Use the following fact: n( S ) + n( S ´ ) = n( U) 9 + 13 = n( U) n( U) = 22 • Recall that all 4 basic regions must add up to n( U). • Thus: Basic region IV = 22 – ( 4 + 5 + 6 ) = 7 With this information we then can fill in the Venn Diagram.

7. Exercise 16 Venn Diagram U S T 5 4 6 7

8. 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)

9. Exercise 23 Venn Diagram U Basic region IV = 70 – ( 30 + 5 + 10 ) = 25 S T 5 30 10 25

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 }

11. 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 business n( 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)

12. Exercise 31 Venn Diagram U M B Basic Region VIII = 35 – ( 2 + 3 + 1 + 11 + 6 + 4 + 6 ) = 2 3 1 2 11 4 6 F 2 6