Question

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.
a) how many of the cities surveyed had only a professional sports team? (simplify your answer.)
b) how many of the cities surveyed had a professional sports team and a symphony, but not a childrens museum? (simplify your answer.)
c) how many of the cities surveyed had a professional sports team or a symphony? (simplify your answer.)
d) how many of the cities surveyed had a professional sports team or a symphony, but not a childrens museum? (simplify your answer.)
e) how many of the cities surveyed had exactly two of the activities? (simplify your 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