1. **Problem Statement:** We need to explain Statement 4: $\left(P \cup A\right)' = 4$ in the context of the Venn diagram for students who enjoy alternative music (A) and pop music (P). 2. **Recall the definitions:** - $P \cup A$ means all students who enjoy either pop or alternative music or both. - $\left(P \cup A\right)'$ means the complement of $P \cup A$, i.e., students who do not enjoy either pop or alternative music. 3. **Given data:** - Total students surveyed: 25 - Number of students who enjoy alternative only: 5 - Number of students who enjoy both (intersection): 3 - Number of students who enjoy pop only: 13 - Sum of these: $5 + 3 + 13 = 21$ 4. **Calculate the complement:** - Students who enjoy neither pop nor alternative music: $$\text{Total} - (P \cup A) = 25 - 21 = 4$$ 5. **Interpretation:** - Statement 4 says $\left(P \cup A\right)' = 4$, which means 4 students do not listen to either type of music. - This matches the calculation and the Venn diagram where the outside region (neither circle) contains 4 students. **Final answer:** Statement 4 is true because the complement of the union of sets $P$ and $A$ contains exactly 4 students who do not enjoy either music type.