Venn Diagram Regions Cc7332

1. **State the problem:** We have a universal set $U$ with $n(U) = 27$ elements and three subsets $A$, $B$, and $C$ with given cardinalities and intersections: $$n(A) = 13, \quad n(B) = 18, \quad n(C) = 17,$$ $$n(A \cap B) = 9, \quad n(A \cap C) = 10, \quad n(B \cap C) = 11,$$ $$n(A \cap B \cap C) = 7.$$ We want to find the number of elements in each of the eight regions of the Venn diagram: - Region I: $A$ only - Region II: $A \cap B$ only (excluding triple intersection) - Region III: $B$ only - Region IV: $A \cap C$ only (excluding triple intersection) - Region V: $A \cap B \cap C$ - Region VI: $B \cap C$ only (excluding triple intersection) - Region VII: $C$ only - Region VIII: outside all three sets 2. **Recall the formula for each region:** - Region V (triple intersection): $$n(A \cap B \cap C) = 7$$ - Region II (in $A$ and $B$ but not $C$): $$n(A \cap B) - n(A \cap B \cap C) = 9 - 7 = 2$$ - Region IV (in $A$ and $C$ but not $B$): $$n(A \cap C) - n(A \cap B \cap C) = 10 - 7 = 3$$ - Region VI (in $B$ and $C$ but not $A$): $$n(B \cap C) - n(A \cap B \cap C) = 11 - 7 = 4$$ 3. **Calculate the elements in only one set:** - Region I (only $A$): $$n(A) - [\text{Region II} + \text{Region IV} + \text{Region V}] = 13 - (2 + 3 + 7) = 13 - 12 = 1$$ - Region III (only $B$): $$n(B) - [\text{Region II} + \text{Region VI} + \text{Region V}] = 18 - (2 + 4 + 7) = 18 - 13 = 5$$ - Region VII (only $C$): $$n(C) - [\text{Region IV} + \text{Region VI} + \text{Region V}] = 17 - (3 + 4 + 7) = 17 - 14 = 3$$ 4. **Calculate elements outside all sets (Region VIII):** Use the formula for the union: $$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$$ Substitute values: $$= 13 + 18 + 17 - 9 - 10 - 11 + 7 = 48 - 30 + 7 = 25$$ Elements outside all sets: $$n(U) - n(A \cup B \cup C) = 27 - 25 = 2$$ 5. **Summary of regions:** - I (only $A$): $1$ - II ($A \cap B$ only): $2$ - III (only $B$): $5$ - IV ($A \cap C$ only): $3$ - V ($A \cap B \cap C$): $7$ - VI ($B \cap C$ only): $4$ - VII (only $C$): $3$ - VIII (outside all): $2$ **Final answer:** $$\boxed{\text{I}=1, \text{II}=2, \text{III}=5, \text{IV}=3, \text{V}=7, \text{VI}=4, \text{VII}=3, \text{VIII}=2}$$