If `A={1,3,5,7,9,11,13,15,17}, B={2,4,.......,18} and N` the set of natural numbers is the universal set, then `A' uu {(A uu B) nn B'}` is (a) ϕ (b)N (c) A (d) B

Options

A. `phi`
B. N
C. A
D. B

Explanation

To solve the problem, we need to find the expression \( A' \cup ( (A \cup B) \cap B' ) \) given the sets \( A \) and \( B \).

### Step 1: Define the Sets
- Let \( A = \{ 1, 3, 5, 7, 9, 11, 13, 15, 17 \} \)
- Let \( B = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18 \} \) (the even numbers from 2 to 18)
- Let \( N \) be the set of natural numbers, which is our universal set.

### Step 2: Find the Complement of Set A
The complement of set \( A \), denoted \( A' \), consists of all elements in the universal set \( N \) that are not in \( A \).
- Thus, \( A' = N - A = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18 \} \)

### Step 3: Find the Union of Sets A and B
Next, we find \( A \cup B \):
- \( A \cup B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 \} \)

### Step 4: Find the Complement of Set B
Now, we find the complement of set \( B \), denoted \( B' \):
- \( B' = N - B = \{ 1, 3, 5, 7, 9, 11, 13, 15, 17 \} \)

### Step 5: Find the Intersection of (A ∪ B) and B'
Now, we compute \( (A \cup B) \cap B' \):
- The intersection consists of elements that are in both \( A \cup B \) and \( B' \):
- \( (A \cup B) \cap B' = \{ 1, 3, 5, 7, 9, 11, 13, 15, 17 \} \)

### Step 6: Find the Union of A' and the Result from Step 5
Finally, we need to find \( A' \cup ( (A \cup B) \cap B' ) \):
- \( A' = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18 \} \)
- \( (A \cup B) \cap B' = \{ 1, 3, 5, 7, 9, 11, 13, 15, 17 \} \)
- Therefore, \( A' \cup ( (A \cup B) \cap B' ) = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 \} \)

### Conclusion
The result is the set of all natural numbers up to 18, which is equal to the set of natural numbers \( N \).

Thus, the answer is \( N \).