https://file.aiquickdraw.com/mathbot/file/1726245775963g0mmcspf.jpg

Let's analyze the Venn diagram and answer the given questions based on the sets \( A \) and \( B \).

### Step 1: Understanding the Diagram
The Venn diagram has the following regions with the number of elements:
- In set \( A \) only (not in \( B \)): 58
- In both sets \( A \) and \( B \): 17
- In set \( B \) only (not in \( A \)): 69
- Outside both sets \( A \) and \( B \) (in the universal set \( U \)): 76

### Step 2: Calculating the Required Values

1. **Total number of elements in \( U \)**:
   \[
   |U| = 58 + 17 + 69 + 76 = 220
   \]

2. **Number of elements in \( A' \)** (complement of \( A \)):
   Elements not in \( A \) include those only in \( B \) and those outside both sets.
   \[
   |A'| = 69 + 76 = 145
   \]

3. **Number of elements in \( A \cap B \)** (intersection of \( A \) and \( B \)):
   \[
   |A \cap B| = 17
   \]

4. **Number of elements in \( A \cap B' \)** (elements in \( A \) but not in \( B \)):
   \[
   |A \cap B'| = 58
   \]

5. **Number of elements in \( A \cup B \)** (union of \( A \) and \( B \)):
   \[
   |A \cup B| = 58 + 17 + 69 = 144
   \]

6. **Number of elements in \( A' \cap B' \)** (neither in \( A \) nor in \( B \)):
   \[
   |A' \cap B'| = 76
   \]

### Summary of Results:
- \( |U| = 220 \)
- \( |A'| = 145 \)
- \( |A \cap B| = 17 \)
- \( |A \cap B'| = 58 \)
- \( |A \cup B| = 144 \)
- \( |A' \cap B'| = 76 \)

Would you like a detailed explanation of any of these calculations, or do you have any specific questions? 

Here are some related questions for further exploration:

1. How would the results change if another element was added to \( A \cap B \)?
2. What is the probability of randomly selecting an element from \( U \) that is only in \( A \)?
3. How do you calculate the symmetric difference between sets \( A \) and \( B \)?
4. What would be the result if an element is added to only set \( B \)?
5. How can Venn diagrams be used to solve problems involving three sets?

**Tip:** When working with Venn diagrams, always double-check to make sure that the counts in all the regions add up to the total in the universal set.

Find the number of elements in each of the following sets based on a Venn diagram: U, A', A ∩ B, A ∩ B', A ∪ B, A' ∩ B'.

Solve a Venn diagram problem with sets A and B, determining the number of elements in their union, intersection, and complements. This problem explores key concepts in set theory, suitable for high school students.

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