Question

a survey of 65 customers was taken at a bookstore regarding the types of books purchased. the survey found that 39 customers purchased mysteries, 31 purchased science fiction, 24 purchased romance novels, 17 purchased mysteries and science fiction, 13 purchased mysteries and romance novels, 10 purchased science fiction and romance novels, and 6 purchased all three types of books.
a) how many of the customers surveyed purchased only science fiction?
b) how many purchased mysteries and science fiction, but not romance novels?
c) how many purchased mysteries or science fiction?
d) how many purchased mysteries or science fiction, but not romance novels?
e) how many purchased exactly two types of books?

a) there were \\(\square\\) customers who purchased only science fiction
(simplify your answer )

Explanation:

Step1: Define the sets

Let \( M \) be the set of customers who purchased mysteries, \( S \) be the set of customers who purchased science fiction, and \( R \) be the set of customers who purchased romance novels. We know \( |M| = 39 \), \( |S| = 31 \), \( |R| = 24 \), \( |M \cap S| = 17 \), \( |M \cap R| = 13 \), \( |S \cap R| = 10 \), and \( |M \cap S \cap R| = 6 \).

Step2: Find the number of customers who purchased only science fiction and one other type

To find the number of customers who purchased only science fiction, we first find the number of customers who purchased science fiction and exactly one other type.

The number of customers who purchased science fiction and mysteries only is \( |M \cap S| - |M \cap S \cap R| = 17 - 6 = 11 \).

The number of customers who purchased science fiction and romance only is \( |S \cap R| - |M \cap S \cap R| = 10 - 6 = 4 \).

Step3: Calculate the number of customers who purchased only science fiction

We use the formula for the number of elements in a set: \( |S| = \) (only \( S \)) \( + \) (only \( S \) and \( M \)) \( + \) (only \( S \) and \( R \)) \( + \) (all three). Let \( x \) be the number of customers who purchased only science fiction. Then:

\( 31 = x + 11 + 4 + 6 \)

\( 31 = x + 21 \)

Subtract 21 from both sides: \( x = 31 - 21 = 10 \)

Answer:

10