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Let's analyze the Venn diagram problem and solve the questions:

Given:
- \( n(A) = 13 \) 
- \( n(B) = 8 \)

From the Venn diagram, the following expressions represent different regions:
- The part exclusive to set \( A \) (outside the overlap): \( 2x + y \)
- The part shared by \( A \) and \( B \) (the intersection): \( x + y \)
- The part exclusive to set \( B \): \( 3y - x \)
- The part outside both sets \( A \) and \( B \) (in the universal set \( U \)): \( 2x - y \)

Now, we solve the specific questions:

### a) \( n(A \cup B) \)
The union of sets \( A \) and \( B \) includes all the regions inside \( A \) and \( B \), i.e., all parts except the region outside the sets:
\[
n(A \cup B) = (2x + y) + (x + y) + (3y - x) = 2x + y + x + y + 3y - x = 2x + x - x + y + y + 3y = 2x + 5y
\]

### b) \( n(U) \)
The total number of elements in the universal set \( U \) includes all regions, i.e., the sum of all four regions:
\[
n(U) = (2x + y) + (x + y) + (3y - x) + (2x - y)
\]
Simplifying this:
\[
n(U) = 2x + y + x + y + 3y - x + 2x - y = 4x + 3y
\]

### c) \( n(B \setminus A) \)
This is the number of elements exclusive to set \( B \) (the part of \( B \) that is not in \( A \)):
\[
n(B \setminus A) = 3y - x
\]

### d) \( n(A \cap B') \)
This refers to the number of elements in \( A \) that are not in \( B \), i.e., exclusive to set \( A \):
\[
n(A \cap B') = 2x + y
\]

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If you'd like further details or have any questions, let me know! Here are 5 related questions to expand your understanding:

1. How is the principle of inclusion-exclusion used to find \( n(A \cup B) \)?
2. How can you determine \( n(A \cap B) \) from the Venn diagram?
3. How would the expressions change if the sets had more overlap or were disjoint?
4. What is the meaning of \( n(U) \) in set theory, and how is it calculated in different problems?
5. How can Venn diagrams be applied to solve probability problems?

**Tip:** When working with Venn diagrams, always clearly label the exclusive and shared regions to avoid confusion in calculations!

The following Venn diagram shows two sets A and B. If n(A) = 13, n(B) = 8, then find:

a) n(A ∪ B)
b) n(U)
c) n(B \ A)
d) n(A ∩ B')

Learn how to solve a Venn diagram problem involving sets A and B with given cardinalities n(A) = 13 and n(B) = 8. This solution explores the union, intersection, and complement of sets using the principles of set theory. Suitable for students in Grades 9-11.

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