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 of A and B) - $A \cap B$ (intersection of A and B) - $A - B$ (elements in A not in B) - $B - A$ (elements in B not in A) Then explain what each represents and identify students taking both subjects. 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 elements common to 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 each set:** - $A \cup B = \{Anna, Ben, Carl, Dana, Ethan, Faith, George, Hannah\}$ - $A \cap B = \{Carl, Dana, Ethan\}$ - $A - B = \{Anna, Ben, Faith\}$ - $B - A = \{George, Hannah\}$ 4. **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. 5. **Answer the question about students taking both subjects:** Students taking both Programming and Networking are those in $A \cap B$: $$\{Carl, Dana, Ethan\}$$ **Final answers:** - $A \cup B = \{Anna, Ben, Carl, Dana, Ethan, Faith, George, Hannah\}$ - $A \cap B = \{Carl, Dana, Ethan\}$ - $A - B = \{Anna, Ben, Faith\}$ - $B - A = \{George, Hannah\}$