In a class with 35 students, a survey was conducted on preferences for mathematics and literature, and it was found that:

- 7 men like mathematics;
- 6 men like literature;
- 5 men and 8 women said they do not like either;
- there are 16 men in the class;
- 5 students like both; and
- 11 students like only mathematics.

a) How many men like both mathematics and literature?

b) How many women like only literature?
Solution

Let $H$ be the set of men and $U$ the total set of people, so $U-H$ is the set of women. In addition, consider the sets Mat and $L$ of people who like mathematics and literature, respectively. If $x$ represents the number of men who like both mathematics and literature and $y$ the number of women who like only literature, we have the following diagram:

![](https://cdn.mathpix.com/cropped/2024_05_01_4fd1b602786ad9981479g-05.jpg?height=457&width=766&top_left_y=837&top_left_x=717)

a) Since there are 16 men in the room,

$$
\begin{aligned}
5+7-x+x+6-x & =16 \\
x & =2
\end{aligned}
$$

b) Since the room is composed of 35 students

$$
\begin{aligned}
|H|+|U-H| & =35 \\
16+4+x+5-x+y+8 & =35 \\
y & =2
\end{aligned}
$$