1. **State the problem:** Find the values of $n(A)$, $n(B)$, $n(A \cap B)$, $n(E)$, $n(A \cup B)$, and $n(B' \cap A)$ from the given Venn diagram. 2. **Given data from the Venn diagram:** - Elements in $A$ only: 3, 6, 15 - Elements in $B$ only: 4, 8, 20 - Elements in $A \cap B$: 12, 24, 16 - Elements outside $A$ and $B$ (in universal set $E$): 2, 10, 5, 7, 11, 13, 17 3. **Calculate $n(A)$:** $n(A)$ is the number of elements in set $A$, including those in the intersection with $B$. $$n(A) = |A \text{ only}| + |A \cap B| = 3 + 6 + 15 + 12 + 24 + 16 = 3 + 6 + 15 + 12 + 24 + 16 = 76$$ 4. **Calculate $n(B)$:** $n(B)$ is the number of elements in set $B$, including those in the intersection with $A$. $$n(B) = |B \text{ only}| + |A \cap B| = 4 + 8 + 20 + 12 + 24 + 16 = 4 + 8 + 20 + 12 + 24 + 16 = 84$$ 5. **Calculate $n(A \cap B)$:** $n(A \cap B)$ is the number of elements common to both $A$ and $B$. $$n(A \cap B) = 12 + 24 + 16 = 52$$ 6. **Calculate $n(E)$:** $n(E)$ is the total number of elements in the universal set, including all elements in $A$, $B$, and outside both. $$n(E) = n(A \text{ only}) + n(B \text{ only}) + n(A \cap B) + \text{outside elements}$$ $$= (3 + 6 + 15) + (4 + 8 + 20) + (12 + 24 + 16) + (2 + 10 + 5 + 7 + 11 + 13 + 17)$$ $$= 24 + 32 + 52 + 65 = 173$$ 7. **Calculate $n(A \cup B)$:** $n(A \cup B)$ is the number of elements in $A$ or $B$ or both. $$n(A \cup B) = n(A) + n(B) - n(A \cap B) = 76 + 84 - 52 = 108$$ 8. **Calculate $n(B' \cap A)$:** $B'$ is the complement of $B$, so $B' \cap A$ are elements in $A$ but not in $B$. $$n(B' \cap A) = n(A) - n(A \cap B) = 76 - 52 = 24$$ **Final answers:** - $n(A) = 76$ - $n(B) = 84$ - $n(A \cap B) = 52$ - $n(E) = 173$ - $n(A \cup B) = 108$ - $n(B' \cap A) = 24$