thirty - five cities were researched to determine whether they had a professional sports team, a symphony…

thirty - five cities were researched to determine whether they had a professional sports team, a symphony, or a childrens museum. of these cities, 14 had a professional sports team, 18 had a symphony, 19 had a childrens museum, 7 had a professional sports team and a symphony, 11 had a professional sports team and a childrens museum, 5 had all three activities. complete parts a) through e) below.\na) how many of the cities surveyed had only a professional sports team? (simplify your answer.)\nb) how many of the cities surveyed had a professional sports team and a symphony, but not a childrens museum? (simplify your answer.)\nc) how many of the cities surveyed had a professional sports team or a symphony? (simplify your answer.)\nd) how many of the cities surveyed had a professional sports team or a symphony, but not a childrens museum? (simplify your answer.)\ne) how many of the cities surveyed had exactly two of the activities? (simplify your answer.)

thirty - five cities were researched to determine whether they had a professional sports team, a symphony, or a childrens museum. of these cities, 14 had a professional sports team, 18 had a symphony, 19 had a childrens museum, 7 had a professional sports team and a symphony, 11 had a professional sports team and a childrens museum, 5 had all three activities. complete parts a) through e) below.\na) how many of the cities surveyed had only a professional sports team? (simplify your answer.)\nb) how many of the cities surveyed had a professional sports team and a symphony, but not a childrens museum? (simplify your answer.)\nc) how many of the cities surveyed had a professional sports team or a symphony? (simplify your answer.)\nd) how many of the cities surveyed had a professional sports team or a symphony, but not a childrens museum? (simplify your answer.)\ne) how many of the cities surveyed had exactly two of the activities? (simplify your answer.)

Answer

Explanation:

Let (S) be the set of cities with a symphony, (P) be the set of cities with a professional sports team, and (M) be the set of cities with a children's museum. We know (n(S) = 19), (n(P)=14), (n(M) = 18), (n(P\cap S)=7), (n(P\cap M)=11), (n(S\cap M)=10), and (n(P\cap S\cap M)=5).

Step1: Use the principle of inclusion - exclusion

The formula for (n(A\cup B\cup C)) is (n(A)+n(B)+n(C)-n(A\cap B)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)). We can also use Venn - diagram based reasoning for subsets.

Step2: Calculate part a

To find the number of cities with only a professional sports team, we use the formula (n(P)-n(P\cap S)-n(P\cap M)+n(P\cap S\cap M)). [n(P)-n(P\cap S)-n(P\cap M)+n(P\cap S\cap M)=14 - 7-11 + 5=1]

Step3: Calculate part b

To find the number of cities with a professional sports team and a symphony but not a children's museum, we use (n(P\cap S)-n(P\cap S\cap M)). [n(P\cap S)-n(P\cap S\cap M)=7 - 5=2]

Step4: Calculate part c

To find the number of cities with a professional sports team or a symphony, we use (n(P)+n(S)-n(P\cap S)). [n(P)+n(S)-n(P\cap S)=14 + 19-7=26]

Step5: Calculate part d

To find the number of cities with a professional sports team or a symphony but not a children's museum, we first find (n((P\cup S)\cap\overline{M})). We know (n(P\cup S) = 26). Also, (n((P\cup S)\cap M)=n((P\cap M)\cup(S\cap M))=n(P\cap M)+n(S\cap M)-n(P\cap S\cap M)=11 + 10-5 = 16). So (n((P\cup S)\cap\overline{M})=n(P\cup S)-n((P\cup S)\cap M)=26-16 = 10)

Step6: Calculate part e

To find the number of cities with exactly two of the activities, we use ((n(P\cap S)-n(P\cap S\cap M))+(n(P\cap M)-n(P\cap S\cap M))+(n(S\cap M)-n(P\cap S\cap M))). [=(7 - 5)+(11 - 5)+(10 - 5)=2 + 6+5=13]

Answer:

a. 1 b. 2 c. 26 d. 10 e. 13