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?
4 (simplify your answer.)

b) how many of the cities surveyed had a professional sports team and a symphony, but not a children’s museum?
7 (simplify your answer.)

c) how many of the cities surveyed had a professional sports team or a symphony?
23 (simplify your answer.)

d) how many of the cities surveyed had a professional sports team or a symphony, but not a children’s museum?
□ (simplify your answer.)

Explanation:

Step1: Identify relevant sets

Let \( S \) = professional sports team, \( Sy \) = symphony, \( M \) = children’s museum. We know:

• \( n(S) = 16 \), \( n(Sy) = 17 \), \( n(M) = 14 \)
• \( n(S \cap Sy) = 10 \), \( n(S \cap M) = 5 \), \( n(Sy \cap M) = 7 \)
• \( n(S \cap Sy \cap M) = 3 \)

Step2: Find only \( S \), only \( Sy \), only \( S \cap Sy \) without \( M \)

• Only \( S \): \( n(S) - n(S \cap Sy) - n(S \cap M) + n(S \cap Sy \cap M) = 16 - 10 - 5 + 3 = 4 \) (from part a)
• Only \( Sy \): \( n(Sy) - n(S \cap Sy) - n(Sy \cap M) + n(S \cap Sy \cap M) = 17 - 10 - 7 + 3 = 3 \)
• \( S \cap Sy \) but not \( M \): \( n(S \cap Sy) - n(S \cap Sy \cap M) = 10 - 3 = 7 \) (from part b)

Step3: Calculate \( S \) or \( Sy \) but not \( M \)

This is (only \( S \)) + (only \( Sy \)) + (\( S \cap Sy \) but not \( M \)):
\( 4 + 3 + 7 = 14 \)

Answer:

14