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 $A$: $n(A \cap B') = 2x + y$

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!