Set Theory

Let's list the positive even numbers:

2, 4, 6, 8, 10, 12, ...

Now, we only want the ones that are less than 10:

2, 4, 6, 8

So, set $A$ is $2,4,6,8$.

Let $F$ be the set of students who like fantasy books, so $|F|=15$.

Let $M$ be the set of students who like mystery books, so $|M|=10$.

The number of students who like both fantasy $and$ mystery books is $6$. This is the intersection of the two sets, $|F\cap M|=6$.

To find the number of students who like fantasy $or$ mystery books ($|F\cup M|$), we use the formula:

$|F\cup M|=|F|+|M|-|F\cap M|$

Substitute the given values:

$|F\cup M|=15+10-6$

$|F\cup M|=25-6$

$|F\cup M|=19$

So, 19 students like to read fantasy or mystery books.

Based on the description:

- $7$ children own only cats (this is part of circle $C$ but outside the overlap).

- $4$ children are in the overlapping region between $C$ and $D$. This represents children who own both cats $and$ dogs.

- $9$ children own only dogs (this is part of circle $D$ but outside the overlap).

Therefore, $4$ children own both cats and dogs.

Let $B$ be the set of students who baked cookies. So, $|B|=12$.

We want to find the number of students who did $not$ bake cookies, which is the complement of set $B$ (denoted as $B^{\prime }$ or $B^c$).

To find the number of students who did not bake cookies, we subtract the number of students who did bake cookies from the total number of students:

Number of students who did not bake cookies = Total students - Number of students who baked cookies

Number of students who did not bake cookies = $25-12$

Number of students who did not bake cookies = $13$

So, 13 students did not bake cookies that day.