https://file.aiquickdraw.com/mathbot/file/1726398341736zh566ewa.jpg

The Venn diagram in the image provides information for two sets, \( A \) and \( B \), with the values within each section.

Let's break down the elements:

- The value 29 represents the number of elements in \( A \) that are only in \( A \) (but not in \( B \)).
- The value 35 represents the number of elements in \( B \) that are only in \( B \) (but not in \( A \)).
- The value 28 represents the number of elements that are in both \( A \) and \( B \).
- The value 25 represents the number of elements that are in neither \( A \) nor \( B \).

Now, let's answer each part:

1. \( n(A) \) - The total number of elements in \( A \) (including those in \( A \cap B \)):
   \[
   n(A) = 29 + 28 = 57
   \]

2. \( n(A \cup B) \) - The total number of elements in either \( A \), \( B \), or both:
   \[
   n(A \cup B) = 29 + 28 + 35 = 92
   \]

3. \( n(B) \) - The total number of elements in \( B \) (including those in \( A \cap B \)):
   \[
   n(B) = 35 + 28 = 63
   \]

4. \( n(A \cap B) \) - The number of elements in both \( A \) and \( B \):
   \[
   n(A \cap B) = 28
   \]

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. What is the value of \( n(A^c \cap B) \)?
2. How do you calculate \( n(A^c \cup B) \)?
3. What is the total number of elements in the universal set?
4. How can you find \( n(A^c \cap B^c) \)?
5. What is the formula for the union of three sets, if a third set \( C \) is introduced?

**Tip:** When working with Venn diagrams, always make sure to include the intersection and exclusive regions carefully to avoid double-counting!

Consider the Venn Diagram. Find n(A), n(A ∪ B), n(B), and n(A ∩ B) using the provided diagram.

Learn how to use a Venn diagram to calculate n(A), n(A ∪ B), n(B), and n(A ∩ B). This example explores key set theory concepts such as union, intersection, and exclusive elements.

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