Suppose a list A contains the 30 students in a mathematics class, and a list B contains the 35 students in an English class, and suppose there are 20 names on both lists. Find the number of students: 1. only on list A, 2. only on list B, 3. on list A or B (or both), 4. on exactly one list.

To solve this problem, we can use the principle of set theory. Let's define the following: - Let \( |A| \) be the number of students in list A (mathematics class). - Let \( |B| \) be the number of students in list B (English class). - Let \( |A \cap B| \) be the number of students who are on both lists. From the problem, we have: - \( |A| = 30 \) - \( |B| = 35 \) - \( |A \cap B| = 20 \) Now, we can find the required values step by step: 1. **Number of students only on list A**: \[ |A \text{ only}| = |A| - |A \cap B| = 30 - 20 = 10 \] 2. **Number of students only on list B**: \[ |B \text{ only}| = |B| - |A \cap B| = 35 - 20 = 15 \] 3. **Number of students on list A or B (or both)**: This can be calculated using the formula for the union of two sets: \[ |A \cup B| = |A| + |B| - |A \cap B| = 30 + 35 - 20 = 45 \] 4. **Number of students on exactly one list**: This is the sum of the students only on list A and only on list B: \[ |A \text{ only}| + |B \text{ only}| = 10 + 15 = 25 \] Now, let's summarize the results: 1. Students only on list A: **10** 2. Students only on list B: **15** 3. Students on list A or B (or both): **45** 4. Students on exactly one list: **25**