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 will first define the sets A, B, and C based on the given definitions, and then analyze the relationships between these sets.

### Step 1: Define the Sets
1. **Set A**:
- Given: \( A = \{ x : x = n, n \in \mathbb{N} \} \)
- This means \( A \) contains all natural numbers:
\[ A = \{ 1, 2, 3, 4, 5, \ldots \} \]

2. **Set B**:
- Given: \( B = \{ x : x = 2n, n \in \mathbb{N} \} \)
- This means \( B \) contains all even natural numbers:
\[ B = \{ 2, 4, 6, 8, 10, \ldots \} \]

3. **Set C**:
- Given: \( C = \{ x : x = 4n, n \in \mathbb{N} \} \)
- This means \( C \) contains all multiples of 4:
\[ C = \{ 4, 8, 12, 16, 20, \ldots \} \]

### Step 2: Find \( A \cap B \) (Intersection of A and B)
- The intersection \( A \cap B \) consists of elements that are common to both sets A and B.
- Since \( B \) contains all even numbers, the common elements are:
\[ A \cap B = \{ 2, 4, 6, 8, 10, \ldots \} = B \]

### Step 3: Find \( A \cup B \) (Union of A and B)
- The union \( A \cup B \) consists of all elements that are in either set A or set B.
- Since \( A \) contains all natural numbers and \( B \) contains all even numbers, the union is:
\[ A \cup B = A = \{ 1, 2, 3, 4, 5, 6, 7, 8, \ldots \} \]

### Step 4: Verify \( A \cap B = B \cup C \)
- We already established that \( A \cap B = B \).
- Now we need to find \( B \cup C \):
- Since \( B \) contains all even numbers and \( C \) contains multiples of 4, the union is:
\[ B \cup C = \{ 2, 4, 6, 8, 10, 12, 14, 16, \ldots \} = B \]
- Hence, \( A \cap B = B \cup C \).

### Step 5: Verify \( B \cup C = B \)
- We already found that \( B \cup C = B \) since all elements of \( C \) are included in \( B \).

### Conclusion
All options provided in the question are correct:
1. \( A \cap B = B \)
2. \( A \cup B = A \)
3. \( A \cap B = B \cup C \)
4. \( B \cup C = B \)