1. **State the problem:** We are given a Venn diagram with three sets $A$, $B$, and $C$ and their respective cardinalities in each region. We need to find the cardinality of the union $A \cup C$. 2. **Recall the formula for the union of two sets:** $$|A \cup C| = |A| + |C| - |A \cap C|$$ This formula accounts for the overlap between $A$ and $C$ so we don't double-count elements. 3. **Identify the values from the Venn diagram:** - $A$ only: 10 - $A \cap B$ only: 3 (not relevant for $A \cup C$ directly) - $B$ only: 11 (not relevant) - $A \cap C$ only: 2 - $A \cap B \cap C$: 5 - $B \cap C$ only: 7 (not relevant) - $C$ only: 8 - Outside all sets: 4 (not relevant) 4. **Calculate $|A|$:** $$|A| = 10 + 3 + 2 + 5 = 20$$ 5. **Calculate $|C|$:** $$|C| = 2 + 5 + 7 + 8 = 22$$ 6. **Calculate $|A \cap C|$:** $$|A \cap C| = 2 + 5 = 7$$ 7. **Apply the union formula:** $$|A \cup C| = 20 + 22 - 7 = 35$$ **Final answer:** $$\boxed{35}$$