1. **State the problem:** We are given two sets $A$ and $B$ with the following information: - $n(A) = 30$ - $n(B) = 70$ - $n(A \cup B) = 80$ - $n(U) = 200$ (where $U$ is the universal set) We need to find the number of elements in the four disjoint subsets of the Venn diagram: $n(A \cap B')$, $n(A \cap B)$, $n(A' \cap B)$, and $n(A' \cap B')$. 2. **Recall the formulas and rules:** - The union formula: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ - The universal set is the total number of elements: $$n(U) = n(A \cup B) + n(A' \cap B')$$ - The subsets are disjoint and cover the entire universal set: $$n(U) = n(A \cap B') + n(A \cap B) + n(A' \cap B) + n(A' \cap B')$$ 3. **Find $n(A \cap B)$:** Using the union formula: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ Substitute known values: $$80 = 30 + 70 - n(A \cap B)$$ $$80 = 100 - n(A \cap B)$$ Rearranged: $$n(A \cap B) = 100 - 80 = 20$$ 4. **Find $n(A \cap B')$:** Given in the problem: $$n(A \cap B') = 10$$ 5. **Find $n(A' \cap B)$:** Since $n(B) = n(A \cap B) + n(A' \cap B)$, we have: $$70 = 20 + n(A' \cap B)$$ So: $$n(A' \cap B) = 70 - 20 = 50$$ 6. **Find $n(A' \cap B')$:** Using the universal set total: $$n(U) = n(A \cap B') + n(A \cap B) + n(A' \cap B) + n(A' \cap B')$$ Substitute known values: $$200 = 10 + 20 + 50 + n(A' \cap B')$$ $$200 = 80 + n(A' \cap B')$$ Rearranged: $$n(A' \cap B') = 200 - 80 = 120$$ **Final answers:** - $n(A \cap B') = 10$ - $n(A \cap B) = 20$ - $n(A' \cap B) = 50$ - $n(A' \cap B') = 120$