1. **Problem Statement:** We have a class of 25 students surveyed about their music preferences: alternative (A) or pop (P). The numbers are: - Only A: 5 - Both A and P: 3 - Only P: 13 - Neither (outside A and P): 4 We need to analyze four statements about these sets and determine which are true using Venn diagrams. 2. **Recall Venn Diagram Basics:** - $A \subseteq P'$ means all elements of $A$ are outside $P$ (no overlap). - $P \cup A$ is the union: all students who like either or both. - $P \cap A$ is the intersection: students who like both. - $(P \cup A)'$ is the complement of the union: students who like neither. 3. **Calculate values:** - $|A| = 5 + 3 = 8$ - $|P| = 13 + 3 = 16$ - $|P \cup A| = 5 + 3 + 13 = 21$ - $|P \cap A| = 3$ - $|(P \cup A)'| = 25 - 21 = 4$ 4. **Analyze each statement:** **Statement 1: $A \subseteq P'$** - This means no overlap between $A$ and $P$. - But $|P \cap A| = 3 \neq 0$, so this is false. **Statement 2: $P \cup A = 21$** - True, as calculated. **Statement 3: $P \cap A = 21$** - False, intersection is 3, not 21. **Statement 4: $(P \cup A)' = 4$** - True, complement of union is 4. 5. **Venn Diagram Justifications:** - For Statement 2, shade the entire union region (5 + 3 + 13 = 21). - For Statement 4, shade the outside region (4 students outside both circles). 6. **Summary:** - True statements: 2 and 4. - False statements: 1 and 3.