`B= {x:x in N and x is an even prime number }`

Answer

To solve the problem, we need to identify the set \( B \) defined as follows:

\[ B = \{ x : x \in \mathbb{N} \text{ and } x \text{ is an even prime number} \} \]

### Step 1: Understand the terms
- **Natural Numbers (\(\mathbb{N}\))**: The set of natural numbers typically includes all positive integers: \( 1, 2, 3, 4, \ldots \).
- **Even Numbers**: These are numbers that are divisible by 2.
- **Prime Numbers**: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

### Step 2: Identify even prime numbers
We need to find numbers that are both even and prime.

1. The only even natural number is \( 2 \) (since all other even numbers can be divided by 2 and hence have at least three divisors: 1, 2, and the number itself).
2. Check if \( 2 \) is a prime number:
- The divisors of \( 2 \) are \( 1 \) and \( 2 \).
- Since it has only two divisors, it is indeed a prime number.

### Step 3: Form the set \( B \)
Since \( 2 \) is the only number that meets the criteria of being both an even number and a prime number, we can conclude that:

\[ B = \{ 2 \} \]

### Step 4: Determine if the set is finite or infinite
- The set \( B \) contains only one element, which is \( 2 \).
- A set that contains a finite number of elements is called a finite set.

Thus, \( B \) is a finite set.

### Final Answer
The set \( B \) is finite and contains the single element \( 2 \).

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