Ms. Teller surveyed the 35 students in her class about their families. 29 students have a brother or a sister, 15 students have a brother, and 9 students have both a brother and a sister. How many students have only a sister?

We can solve this problem using the principle of set theory, particularly by applying the inclusion-exclusion principle.

Let's define the following sets:
- Let \( B \) be the set of students who have a brother.
- Let \( S \) be the set of students who have a sister.

We are given the following information:
- \( |B \cup S| = 29 \) (29 students have a brother or a sister),
- \( |B| = 15 \) (15 students have a brother),
- \( |B \cap S| = 9 \) (9 students have both a brother and a sister).

We are asked to find how many students have only a sister, i.e., how many students are in \( S \) but not in \( B \) (denoted as \( |S - B| \)).

### Step 1: Use the inclusion-exclusion principle.
The inclusion-exclusion principle for two sets states:

\[
|B \cup S| = |B| + |S| - |B \cap S|
\]

Substitute the known values into this equation:

\[
29 = 15 + |S| - 9
\]

Simplify the equation:

\[
29 = 6 + |S|
\]

\[
|S| = 23
\]

So, 23 students have a sister.

### Step 2: Calculate the number of students who have only a sister.
The number of students who have only a sister is the total number of students who have a sister minus those who have both a brother and a sister:

\[
|S - B| = |S| - |B \cap S| = 23 - 9 = 14
\]

### Final Answer:
There are 14 students who have only a sister.

Would you like further details on the process or have any questions?

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**Related Questions:**
1. How many students have only a brother?
2. How many students do not have any siblings?
3. How can we use Venn diagrams to visualize this solution?
4. If 3 more students had both a brother and a sister, how would the answer change?
5. How does inclusion-exclusion work with three or more sets?

**Tip:** Drawing a Venn diagram can help visualize set problems and make inclusion-exclusion calculations more intuitive.

Solve a set theory problem where Ms. Teller surveyed her class about siblings. Learn how to apply the inclusion-exclusion principle to find how many students have only a sister, given data about those with brothers or both siblings. Suitable for students in Grades 7-9.

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