The set `(AuuB^(prime))^'uu(BnnC)` is equal to `A 'uuBuuC` b. `A 'uuB` c. `A 'uuC '` d. `A 'nnB`

Options

A. `A' uu B uu C`
B. `A' uu B`
C. `A' uu C'`
D. `A' nn B`

Explanation

To solve the problem, we need to analyze the expression given and determine which of the provided options is equivalent to it.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We have the expression \((A \cup B')' \cup (B \cap C)\). We need to simplify this expression.

2. **Finding \(B'\)**:
- The complement of set \(B\) (denoted as \(B'\)) is found by taking the universal set and removing the elements of \(B\).
- Let's assume the universal set \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) and \(B = \{6, 7, 8\}\).
- Thus, \(B' = U - B = \{1, 2, 3, 4, 5, 9, 10\}\).

3. **Finding \(A \cup B'\)**:
- Let’s assume \(A = \{1, 2, 3, 4\}\).
- Therefore, \(A \cup B' = \{1, 2, 3, 4\} \cup \{1, 2, 3, 4, 5, 9, 10\} = \{1, 2, 3, 4, 5, 9, 10\}\).

4. **Finding \((A \cup B')'\)**:
- The complement of \(A \cup B'\) is \(U - (A \cup B')\).
- Thus, \((A \cup B')' = U - \{1, 2, 3, 4, 5, 9, 10\} = \{6, 7, 8\}\).

5. **Finding \(B \cap C\)**:
- Assuming \(C = \{9, 10\}\), we find \(B \cap C = \{6, 7, 8\} \cap \{9, 10\} = \emptyset\).

6. **Combining the Results**:
- Now we combine the results: \((A \cup B')' \cup (B \cap C) = \{6, 7, 8\} \cup \emptyset = \{6, 7, 8\}\).

7. **Comparing with Options**:
- Now we need to check which of the options matches \(\{6, 7, 8\}\).
- Option A: \(A' \cup B \cup C\)
- Option B: \(A' \cup B\)
- Option C: \(A' \cup C'\)
- Option D: \(A' \cap B\)

- We calculate \(A' = \{5, 6, 7, 8, 9, 10\}\) and check each option:
- **Option A**: \(A' \cup B \cup C = \{5, 6, 7, 8, 9, 10\} \cup \{6, 7, 8\} \cup \{9, 10\} = \{5, 6, 7, 8, 9, 10\}\) (not equal).
- **Option B**: \(A' \cup B = \{5, 6, 7, 8, 9, 10\} \cup \{6, 7, 8\} = \{5, 6, 7, 8, 9, 10\}\) (not equal).
- **Option C**: \(A' \cup C' = \{5, 6, 7, 8, 9, 10\} \cup \{1, 2, 3, 4, 5\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) (not equal).
- **Option D**: \(A' \cap B = \{5, 6, 7, 8, 9, 10\} \cap \{6, 7, 8\} = \{6, 7, 8\}\) (this is equal).

### Conclusion:
The correct answer is **Option D: \(A' \cap B\)**.