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.