1. The problem involves finding intersections and unions of sets P, Q, X, and Y based on the Venn diagram values. 2. Recall the definitions: - Intersection $A \cap B$ is the set of elements common to both $A$ and $B$. - Union $A \cup B$ is the set of elements in $A$, or $B$, or both. 3. Given values from the Venn diagram: - $P = \{5,6,9,2\}$ - $Q = \{12,6,9,10\}$ - $X = \{4,2,9,10\}$ - $Y = \{8\}$ (outside all circles) 4. Calculate each: **a) $P \cap Q$**: Elements common to $P$ and $Q$ are $6$ and $9$. **b) $P \cup Q$**: Combine all unique elements in $P$ and $Q$: $$\{5,6,9,2,12,10\}$$ **c) $X \cap Q$**: Elements common to $X$ and $Q$ are $9$ and $10$. **d) $X \cup Q$**: Combine all unique elements in $X$ and $Q$: $$\{4,2,9,10,6,12\}$$ **e) $Y \cap Q$**: $Y$ is outside all circles, so no overlap with $Q$: $$\emptyset$$ **f) $P \cap Q \cap X$**: Elements common to all three sets is $9$. 5. Check your probability statements: - $0.7 = 0.5 + 0.2 - P(A \cap B)$ implies $P(A \cap B) = 0$ is correct. - $P(A \cup B) = 0.7 + 0.2 - 0.15 = 0.75$ is correct. - $0.9 = 0.3 + P(B) - 0$ implies $P(B) = 0.6$ is correct. Final answers: - a) $\{6,9\}$ - b) $\{2,5,6,9,10,12\}$ - c) $\{9,10\}$ - d) $\{2,4,6,9,10,12\}$ - e) $\emptyset$ - f) $\{9\}$