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?
□ (simplify your answer.)

Explanation:

Step1: Recall the principle of inclusion - exclusion for two sets. The formula for \(|A \cup B|\) is \(|A|+|B|-|A \cap B|\), where \(A\) is the set of cities with a professional sports team and \(B\) is the set of cities with a symphony.

Let \(|A| = 16\) (number of cities with a professional sports team), \(|B|=17\) (number of cities with a symphony), and \(|A\cap B| = 10\) (number of cities with both a professional sports team and a symphony).

Step2: Apply the inclusion - exclusion formula.

Substitute the values into the formula: \(|A\cup B|=|A| + |B|-|A\cap B|\)
\(|A\cup B|=16 + 17-10\)
First, calculate \(16 + 17=33\), then \(33-10 = 23\).

Answer:

23