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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 = 12 + 14 – n( S ∩ T ) n( S ∩ T ) = 8 Exercise 13 Venn Diagram U S 4 T 8 6 2 Basic region IV = 20 – ( 4 + 8 + 6 ) = 2 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 ) 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. Exercise 16 Venn Diagram U S 4 7 T 5 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” “35 students like rock music” “15 students like hip-hop” “5 liked both” n(U) = 70 n(S) = 35 n(T) = 15 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 U S 30 T 5 10 25 Basic region IV = 70 – ( 30 + 5 + 10 ) = 25 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” “22 are male students” “19 are business students” “27 are first-year students” “17 are male first-year” “15 are first-year business “14 are male business” “11 are male first-year business” n(U) = 35 n(M) = 22 n(B) = 19 n(F) = 27 n( M ∩ F ) = 17 n( B ∩ F ) = 15 n( M ∩ B ) = 14 n( M ∩ B ∩ F ) = 11 ** ( ** means that this is a basic region in the Venn Diagram) Exercise 31 Venn Diagram M 3 2 1 11 4 6 2 B F 6 Basic Region VIII = 35 – ( 2 + 3 + 1 + 11 + 6 + 4 + 6 ) = 2 U