Question: Directions Read each problem carefully. Show all necessary solutions, illustrations, and explanations. Use proper mathematical notation. Problem 1. Student Enrollment Analysis A = Students enrolled in Programming = ${Anna, Ben, Carl, Dana, Ethan, Faith}$ B = Students enrolled in Networking = ${Carl, Dana, George, Hannah, Ethan}$ Questions: a. Determine $A \cup B$, $A \cap B$, $A - B$, and $B - A$. b. Explain what each answer represents. c. Which students are taking both subjects?

1. **State the problem:** We have two sets: $$A = \{Anna, Ben, Carl, Dana, Ethan, Faith\}$$ $$B = \{Carl, Dana, George, Hannah, Ethan\}$$ We need to find: - $A \cup B$ (union) - $A \cap B$ (intersection) - $A - B$ (difference) - $B - A$ (difference) 2. **Recall set operation definitions:** - $A \cup B$ is the set of all elements in $A$ or $B$ or both. - $A \cap B$ is the set of all elements in both $A$ and $B$. - $A - B$ is the set of elements in $A$ but not in $B$. - $B - A$ is the set of elements in $B$ but not in $A$. 3. **Calculate $A \cup B$:** Combine all unique students from both sets: $$A \cup B = \{Anna, Ben, Carl, Dana, Ethan, Faith, George, Hannah\}$$ 4. **Calculate $A \cap B$:** Find students in both $A$ and $B$: $$A \cap B = \{Carl, Dana, Ethan\}$$ 5. **Calculate $A - B$:** Students in $A$ but not in $B$: $$A - B = \{Anna, Ben, Faith\}$$ 6. **Calculate $B - A$:** Students in $B$ but not in $A$: $$B - A = \{George, Hannah\}$$ 7. **Explain what each represents:** - $A \cup B$ represents all students enrolled in Programming or Networking or both. - $A \cap B$ represents students enrolled in both Programming and Networking. - $A - B$ represents students enrolled only in Programming. - $B - A$ represents students enrolled only in Networking. 8. **Students taking both subjects:** From $A \cap B$, the students are: $$\boxed{Carl, Dana, Ethan}$$