https://file.aiquickdraw.com/mathbot/file/1729908656652vm33a9qj.jpg

To solve this problem, let’s analyze the given information and use it to fill in each region in the Venn diagram:

### Given Information:
1. \( n(A) = 25 \): The total number of elements in set \( A \).
2. \( n(B) = 21 \): The total number of elements in set \( B \).
3. \( n(A \cup B) = 29 \): The number of elements in the union of sets \( A \) and \( B \).
4. \( n(A') = 41 \): The number of elements outside both sets \( A \) and \( B \).

### Regions in the Diagram:
- \( x \): Elements in \( A \) only.
- \( y \): Elements in both \( A \) and \( B \) (intersection).
- \( z \): Elements in \( B \) only.
- \( w \): Elements outside both \( A \) and \( B \).

### Solution Steps:
1. **Use the Union Formula**:
   \[
   n(A \cup B) = n(A) + n(B) - n(A \cap B)
   \]
   Substitute the known values:
   \[
   29 = 25 + 21 - n(A \cap B)
   \]
   Solving for \( n(A \cap B) \):
   \[
   n(A \cap B) = 25 + 21 - 29 = 17
   \]
   So, \( y = 17 \) (elements in both \( A \) and \( B \)).

2. **Calculate Elements Only in \( A \)**:
   \[
   x = n(A) - n(A \cap B) = 25 - 17 = 8
   \]

3. **Calculate Elements Only in \( B \)**:
   \[
   z = n(B) - n(A \cap B) = 21 - 17 = 4
   \]

4. **Calculate Elements Outside Both \( A \) and \( B \)**:
   Since \( n(A') = 41 \), we have:
   \[
   w = 41
   \]

### Final Values for Each Region:
- \( x = 8 \)
- \( y = 17 \)
- \( z = 4 \)
- \( w = 41 \)

### Answer Summary:
- The number of elements in region \( x \) is **8**.
- The number of elements in region \( y \) is **17**.
- The number of elements in region \( z \) is **4**.
- The number of elements in region \( w \) is **41**.

Would you like more details or have any questions?

#### Related Questions:
1. How would the solution change if \( n(A \cup B) \) were different?
2. What is the interpretation of each region in practical terms?
3. How would adding an element common to both sets impact \( n(A \cup B) \)?
4. How would the regions be affected if \( n(A') \) were a different value?
5. Can we solve for other unknowns if we change the value of \( n(B) \)?

#### Tip:
For Venn diagrams, always start by calculating intersections, as they help determine elements unique to each set.

Use the given information to fill in the number of elements for each region in the Venn diagram. n(A) = 25, n(B) = 21, n(A U B) = 29, n(A') = 41.

Solve a Venn diagram problem involving sets A and B where n(A) = 25, n(B) = 21, n(A U B) = 29, and n(A') = 41. Learn how to calculate elements in each region of the diagram using set theory concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation