1. **State the problem:** We have a universal set $\xi$ and three sets $C$, $D$, and $E$ with given numbers of elements in each region of their Venn diagram. We need to find: (i) $n(C')$, the number of elements not in set $C$. (ii) $n(C' \cap D')$, the number of elements not in $C$ and not in $D$. 2. **Recall definitions and formulas:** - $C'$ is the complement of $C$, meaning all elements in $\xi$ that are not in $C$. - $C' \cap D'$ is the complement of $C$ intersected with the complement of $D$, meaning elements not in $C$ and not in $D$. 3. **Identify all elements in the universal set $\xi$:** Sum all numbers in the diagram: $$3 + 7 + 6 + 1 + 2 + 12 + 13 + 9 = 53$$ 4. **Find $n(C)$:** Elements in $C$ are those in regions labeled with $C$: - Only $C$: 3 - $C \cap D$: 7 - $C \cap E$: 1 - $C \cap D \cap E$: 2 Sum: $$n(C) = 3 + 7 + 1 + 2 = 13$$ 5. **Calculate $n(C')$:** $$n(C') = n(\xi) - n(C) = 53 - 13 = 40$$ 6. **Find $n(C' \cap D')$:** Elements not in $C$ and not in $D$ are those outside both sets, which are: - Only $E$: 13 - Outside all sets: 9 Sum: $$n(C' \cap D') = 13 + 9 = 22$$ **Final answers:** (i) $n(C') = 40$ (ii) $n(C' \cap D') = 22$