1. **State the problem:** We are given a universal set $U$ with $n(U) = 200$, and two subsets $A$ and $B$ with $n(A) = 20$, $n(B) = 50$, and $n(A \cap B) = 5$. We need to find the number of elements in each of the four disjoint subsets of the Venn diagram: $A \cap B$, $A \cap B'$, $A' \cap B$, and $A' \cap B'$. 2. **Recall the definitions:** - $A \cap B$ is the intersection of $A$ and $B$ (elements in both). - $A \cap B'$ is elements in $A$ but not in $B$. - $A' \cap B$ is elements in $B$ but not in $A$. - $A' \cap B'$ is elements in neither $A$ nor $B$. 3. **Given:** - $n(A) = 20$ - $n(B) = 50$ - $n(A \cap B) = 5$ - $n(U) = 200$ 4. **Find $n(A \cap B)$:** This is given as 5. 5. **Find $n(A \cap B')$:** Elements in $A$ but not in $B$. $$n(A \cap B') = n(A) - n(A \cap B) = 20 - 5 = 15$$ 6. **Find $n(A' \cap B)$:** Elements in $B$ but not in $A$. $$n(A' \cap B) = n(B) - n(A \cap B) = 50 - 5 = 45$$ 7. **Find $n(A' \cap B')$:** Elements in neither $A$ nor $B$. Use the formula for the union: $$n(A \cup B) = n(A) + n(B) - n(A \cap B) = 20 + 50 - 5 = 65$$ Since $U$ contains all elements, $$n(A' \cap B') = n(U) - n(A \cup B) = 200 - 65 = 135$$ **Final answers:** - $n(A \cap B) = 5$ - $n(A \cap B') = 15$ - $n(A' \cap B) = 45$ - $n(A' \cap B') = 135$