If A and B are two sets, then `(AuuB')'nn(A'uuB)'` is (i) null set (ii) universal set (iii) `A'` (iv) `B'`

Options

A. null set
B. universal set
C. `A'`
D. `B'`

Answer

To solve the problem `(A ∪ B')' ∩ (A' ∪ B)'`, we will follow the steps of set theory operations, using the properties of complements and intersections.

### Step-by-Step Solution:

1. **Identify the components:**
- Let \( A' \) be the complement of set \( A \).
- Let \( B' \) be the complement of set \( B \).
- We need to find \( (A ∪ B')' \) and \( (A' ∪ B)' \).

2. **Find \( A ∪ B' \):**
- The union of \( A \) and the complement of \( B \) includes all elements that are in \( A \) or not in \( B \).

3. **Find the complement of \( A ∪ B' \):**
- By De Morgan's law, \( (A ∪ B')' = A' ∩ B \).
- This means we are looking for elements that are not in \( A \) and are in \( B \).

4. **Find \( A' ∪ B \):**
- The union of the complement of \( A \) and \( B \) includes all elements that are not in \( A \) or are in \( B \).

5. **Find the complement of \( A' ∪ B \):**
- Again, by De Morgan's law, \( (A' ∪ B)' = A ∩ B' \).
- This means we are looking for elements that are in \( A \) and not in \( B \).

6. **Now, we need to find the intersection:**
- We need to compute \( (A' ∩ B) ∩ (A ∩ B') \).
- This intersection represents elements that are both in \( A' \) and \( B \) and also in \( A \) and \( B' \).

7. **Analyze the intersection:**
- The intersection \( (A' ∩ B) ∩ (A ∩ B') \) consists of elements that are in \( B \) but not in \( A \) and also in \( A \) but not in \( B \).
- There cannot be any element that satisfies both conditions simultaneously, thus the intersection is empty.

8. **Conclusion:**
- Therefore, \( (A ∪ B')' ∩ (A' ∪ B)' = ∅ \) (the null set).

### Final Answer:
The correct option is (i) null set.