`B ={x:x in NN and x^2 =x}`

Answer

To solve the problem, we need to determine the set \( B = \{ x : x \in \mathbb{N} \text{ and } x^2 = x \} \).

### Step-by-Step Solution:

1. **Understand the Definition of the Set**:
The set \( B \) is defined as the collection of all natural numbers \( x \) such that \( x^2 = x \).

2. **Rewrite the Equation**:
The equation \( x^2 = x \) can be rearranged to \( x^2 - x = 0 \).

3. **Factor the Equation**:
We can factor the equation as follows:
\[ x(x - 1) = 0 \]

4. **Find the Solutions**:
The factored equation \( x(x - 1) = 0 \) gives us two solutions:
- \( x = 0 \)
- \( x = 1 \)

5. **Consider the Domain of Natural Numbers**:
The natural numbers \( \mathbb{N} \) typically start from 1 (i.e., \( \mathbb{N} = \{1, 2, 3, \ldots\} \)). Therefore, we only consider the solution \( x = 1 \).

6. **Identify the Set**:
Since \( x = 1 \) is the only solution that belongs to the natural numbers, we can conclude that:
\[ B = \{ 1 \} \]

7. **Conclusion**:
The set \( B \) is not an empty set because it contains the element \( 1 \).

### Final Answer:
The set \( B \) is \( \{ 1 \} \), which is a non-empty set.

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