If n(A∩B) = 50, n(A∩Bc) = 200, n(Ac∩B) = 150, what is n(A∪B)?

Given:

- \( n(A \cap B) = 50 \)
- \( n(A \cap B^c) = 200 \)
- \( n(A^c \cap B) = 150 \)

We need to find \( n(A \cup B) \), the number of elements in the union of sets \( A \) and \( B \).

### Steps to find \( n(A \cup B) \):

1. **Break down the union into disjoint parts:**

   The union of \( A \) and \( B \) can be expressed as the sum of the following disjoint sets:
   - \( A \cap B \) (elements in both \( A \) and \( B \))
   - \( A \cap B^c \) (elements in \( A \) but not in \( B \))
   - \( A^c \cap B \) (elements in \( B \) but not in \( A \))

   Thus:

   \[
   n(A \cup B) = n(A \cap B) + n(A \cap B^c) + n(A^c \cap B)
   \]

2. **Substitute the given values:**

   \[
   n(A \cup B) = 50 + 200 + 150
   \]

3. **Calculate the total:**

   \[
   n(A \cup B) = 400
   \]

So, \( n(A \cup B) = 400 \).

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find \( n(A \cup B) \) using the principle of inclusion-exclusion?
2. What is the significance of the disjoint parts in set operations?
3. How would the calculation change if you had \( n(U) \) (total number of elements in the universal set)?
4. Can you explain how to visualize this problem using a Venn diagram?
5. What are some practical applications of set operations like union and intersection?

**Tip:** When dealing with unions and intersections, breaking down the sets into disjoint parts can simplify the calculation.

Explore how to calculate the union of sets A and B using the principle of inclusion-exclusion. Given n(A ∩ B) = 50, n(A ∩ Bc) = 200, and n(Ac ∩ B) = 150, find n(A ∪ B) with detailed step-by-step solution and explanations.

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