Set Union Fdf7C9

1. The user requests more complex problems involving sets. 2. Let's consider a problem involving set operations and Venn diagrams. 3. Problem: Given sets $A$, $B$, and $C$ with the following properties: - $|A| = 20$ - $|B| = 30$ - $|C| = 25$ - $|A \cap B| = 10$ - $|B \cap C| = 15$ - $|A \cap C| = 5$ - $|A \cap B \cap C| = 3$ Find the number of elements in $A \cup B \cup C$. 4. Formula used: $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$$ 5. Substitute the values: $$|A \cup B \cup C| = 20 + 30 + 25 - 10 - 15 - 5 + 3$$ 6. Simplify step-by-step: $$|A \cup B \cup C| = 75 - 30 + 3$$ $$|A \cup B \cup C| = 48$$ 7. Explanation: To find the total number of elements in the union of three sets, we add the sizes of each set, subtract the sizes of all pairwise intersections to avoid double counting, then add back the size of the triple intersection because it was subtracted three times. 8. Final answer: $$\boxed{48}$$