Question

thirty-three cities were researched to determine whether they had a professional sports team, a symphony, or a children’s museum. of these cities, 16 had a professional sports team, 17 had a symphony, 14 had a children’s museum, 10 had a professional sports team and a symphony, 5 had a professional sports team and a children’s museum, 7 had a symphony and a children’s museum, and 3 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 )

Explanation:

Step1: Identify the formula for only one set

To find the number of cities with only a professional sports team, we use the principle of inclusion - exclusion for three sets. The number of elements in only set \( A \) (professional sports team) is given by \( n(A) - n(A\cap B) - n(A\cap C)+n(A\cap B\cap C) \), where \( A \) is professional sports team, \( B \) is symphony, and \( C \) is children's museum.

Step2: Substitute the values

We know that \( n(A) = 16 \), \( n(A\cap B)=10 \), \( n(A\cap C) = 5 \), and \( n(A\cap B\cap C)=3 \).

Substituting these values into the formula: \( 16- 10 - 5+3 \)

First, calculate \( 16-10=6 \), then \( 6 - 5 = 1 \), and finally \( 1+3=4 \).

Answer:

\( 4 \)