If `A={x : x=n , n in N}`, `B={x : =2n , n in N}`, `c={x : x =4n , n in N}`, then

Options

A. `AnnB={2,4,6,……}`
B. `AuuB={1,2,3,….}`
C. `AnnB=(BuuC)=B`
D. `BuuC=B`

Answer

To solve the problem, we need to analyze the sets A, B, and C given in the question.

1. **Define the Sets**:
- Set A: \( A = \{ x : x = n, n \in \mathbb{N} \} \)
- This means A contains all natural numbers: \( A = \{ 1, 2, 3, 4, 5, 6, \ldots \} \)
- Set B: \( B = \{ x : x = 2n, n \in \mathbb{N} \} \)
- This means B contains all even natural numbers: \( B = \{ 2, 4, 6, 8, 10, 12, \ldots \} \)
- Set C: \( C = \{ x : x = 4n, n \in \mathbb{N} \} \)
- This means C contains all multiples of 4: \( C = \{ 4, 8, 12, 16, 20, 24, \ldots \} \)

2. **Find A ∩ B (Intersection of A and B)**:
- The intersection of A and B includes all elements that are common to both sets.
- Since A contains all natural numbers and B contains all even natural numbers, the intersection will be all even natural numbers:
- \( A \cap B = B = \{ 2, 4, 6, 8, 10, 12, \ldots \} \)

3. **Find A ∪ B (Union of A and B)**:
- The union of A and B includes all elements that are in either A or B.
- Since A contains all natural numbers and B contains all even natural numbers, the union will be all natural numbers:
- \( A \cup B = A = \{ 1, 2, 3, 4, 5, 6, \ldots \} \)

4. **Find B ∪ C (Union of B and C)**:
- The union of B and C includes all elements that are in either B or C.
- B contains all even natural numbers and C contains all multiples of 4. The union will include all even numbers:
- \( B \cup C = \{ 2, 4, 6, 8, 10, 12, 16, 20, \ldots \} \)

5. **Check if A ∩ B = B ∪ C**:
- From the previous steps, we found:
- \( A \cap B = \{ 2, 4, 6, 8, 10, 12, \ldots \} \)
- \( B \cup C = \{ 2, 4, 6, 8, 10, 12, \ldots \} \)
- Therefore, \( A \cap B = B \cup C \).

6. **Check if B ∪ C = B**:
- Since B contains all even natural numbers and C contains multiples of 4 (which are also even), the union will still be all even natural numbers:
- \( B \cup C = B \).

### Summary of Results:
- \( A \cap B = B \)
- \( A \cup B = A \)
- \( B \cup C = B \)
- \( A \cap B = B \cup C \)

### Conclusion:
All the options given in the question are correct.