If `A={x:x=3n,n in N} and B={x:x=4n,n in N}`, then find `A cap B`.

Answer

To solve the problem of finding the intersection of the sets \( A \) and \( B \), we will follow these steps:

### Step 1: Define the sets \( A \) and \( B \)
The set \( A \) is defined as:
\[ A = \{ x : x = 3n, n \in \mathbb{N} \} \]
This means \( A \) consists of all multiples of 3.

The set \( B \) is defined as:
\[ B = \{ x : x = 4n, n \in \mathbb{N} \} \]
This means \( B \) consists of all multiples of 4.

### Step 2: Write the elements of sets \( A \) and \( B \) in roster form
To find the elements of \( A \) and \( B \), we can list the first few multiples of 3 and 4:

- For \( A \):
- When \( n = 1 \): \( 3 \times 1 = 3 \)
- When \( n = 2 \): \( 3 \times 2 = 6 \)
- When \( n = 3 \): \( 3 \times 3 = 9 \)
- When \( n = 4 \): \( 3 \times 4 = 12 \)
- When \( n = 5 \): \( 3 \times 5 = 15 \)
- Continuing this way, we get:
\[ A = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, \ldots \} \]

- For \( B \):
- When \( n = 1 \): \( 4 \times 1 = 4 \)
- When \( n = 2 \): \( 4 \times 2 = 8 \)
- When \( n = 3 \): \( 4 \times 3 = 12 \)
- When \( n = 4 \): \( 4 \times 4 = 16 \)
- When \( n = 5 \): \( 4 \times 5 = 20 \)
- Continuing this way, we get:
\[ B = \{ 4, 8, 12, 16, 20, 24, 28, 32, 36, \ldots \} \]

### Step 3: Find the intersection \( A \cap B \)
The intersection \( A \cap B \) consists of elements that are common to both sets \( A \) and \( B \).

From the lists:
- The common multiples of 3 and 4 are:
- \( 12 \) (present in both)
- \( 24 \) (present in both)
- Continuing this way, we can see that the common multiples will be multiples of \( 12 \).

Thus, the intersection can be expressed as:
\[ A \cap B = \{ x : x = 12k, k \in \mathbb{N} \} \]

### Final Answer
The intersection of sets \( A \) and \( B \) is:
\[ A \cap B = \{ 12, 24, 36, 48, \ldots \} \]
Or in set builder notation:
\[ A \cap B = \{ x : x = 12k, k \in \mathbb{N} \} \]