1. **State the problem:** We are given a Venn diagram with three sets $A$, $B$, and $C$ inside a universal set $U$. We need to find the cardinality of $A'$, the complement of $A$, which means all elements in $U$ that are not in $A$. 2. **Identify the elements in each region:** From the diagram: - $A$ only: 10 - $A \cap B$ only: 3 - $B$ only: 11 - $A \cap C$ only: 2 - $A \cap B \cap C$: 5 - $B \cap C$ only: 7 - Outside $A$, $B$, $C$ (in $U$ only): 4 - $C$ only: 8 3. **Calculate the total number of elements in $U$:** $$|U| = 10 + 3 + 11 + 2 + 5 + 7 + 4 + 8 = 50$$ 4. **Calculate the number of elements in $A$:** Elements in $A$ are those in $A$ only, $A \cap B$ only, $A \cap C$ only, and $A \cap B \cap C$: $$|A| = 10 + 3 + 2 + 5 = 20$$ 5. **Calculate the complement of $A$, denoted $A'$:** By definition, $$|A'| = |U| - |A|$$ Substitute the values: $$|A'| = 50 - 20 = 30$$ **Final answer:** $$\boxed{30}$$