Use the given information to complete the solution of the partially solved Venn diagram. HINT [See Example 4.]
n(A) = 16, n(B) = 11, n(C) = 32, n(S) = 40
x =
y =
z =
w =

We are given the total number of elements in the universal set \( n(S) = 40 \), as well as the number of elements in sets \( A \), \( B \), and \( C \) with:

- \( n(A) = 16 \)
- \( n(B) = 11 \)
- \( n(C) = 32 \)

### Let's define the variables:

- \( x \): Represents the number of elements that are in all three sets \( A \cap B \cap C \).
- \( y \): Represents the number of elements in \( A \cap B \) but not in \( C \).
- \( z \): Represents the number of elements in \( B \cap C \) but not in \( A \).
- \( w \): Represents the number of elements in \( A \cap C \) but not in \( B \).

To solve this problem, we need more details on the relationships between the sets. The hint indicates an example (likely a solved Venn diagram) is provided, but without specific overlap information, we can't solve for \( x \), \( y \), \( z \), and \( w \) directly. These variables typically represent intersections between sets, but we would need the number of elements in the pairwise or three-way intersections to continue.

If additional intersection data or relationships are available, I can continue solving it step-by-step.

Would you like me to explain this further or provide clarification on any part of the Venn diagram setup?

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Here are five questions you can explore next:
1. How are the elements distributed between two sets, \( A \) and \( B \), in a Venn diagram?
2. What is the significance of the intersection \( A \cap B \cap C \) in a Venn diagram?
3. How do the union and intersection operations affect the total number of elements in the sets?
4. How would adding a fourth set \( D \) change the structure of a Venn diagram?
5. What strategies can help when solving more complex Venn diagram problems?

**Tip**: In Venn diagrams, always try to find the intersections first, as they help break down the problem into smaller parts.

Use the given information to complete the solution of the partially solved Venn diagram. HINT [See Example 4.] n(A) = 16, n(B) = 11, n(C) = 32, n(S) = 40 x = y = z = w =

Explore how to solve a Venn diagram problem involving sets A, B, C, and the universal set S. Given data includes n(A) = 16, n(B) = 11, n(C) = 32, and n(S) = 40. Learn about key concepts like set intersections, union, and the Principle of Inclusion-Exclusion. Suitable for students in Grades 9-12.

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