Let `**:NxxNrarrN` be an operation defined as `a**b=a+ab,AAa,binN`
Check if `'**'` is a binary operation.
If yes, find if it is associative too.

Answer

To determine whether the operation `**` defined as \( a ** b = a + ab \) for all \( a, b \in \mathbb{N} \) is a binary operation and whether it is associative, we will follow these steps:

### Step 1: Check if `**` is a binary operation
A binary operation on a set is defined as a function that takes two elements from that set and produces another element from the same set.

1. **Definition of the operation**: Given \( a ** b = a + ab \).
2. **Elements of the set**: Here, \( a \) and \( b \) are both natural numbers (i.e., \( a, b \in \mathbb{N} \)).
3. **Output of the operation**: We need to check if \( a + ab \) is also a natural number.
- Since \( a \) and \( b \) are natural numbers, \( ab \) (the product of two natural numbers) is also a natural number.
- The sum of two natural numbers \( a + ab \) is also a natural number.

Thus, the operation `**` maps \( \mathbb{N} \times \mathbb{N} \) to \( \mathbb{N} \). Therefore, `**` is a binary operation.

### Step 2: Check if the operation is associative
An operation is associative if for all \( a, b, c \in \mathbb{N} \):
\[ (a ** b) ** c = a ** (b ** c) \]

1. **Calculate \( (a ** b) ** c \)**:
- First, compute \( a ** b \):
\[ a ** b = a + ab \]
- Now, substitute this into \( (a ** b) ** c \):
\[ (a ** b) ** c = (a + ab) ** c = (a + ab) + (a + ab)c \]
\[ = a + ab + ac + abc \]

2. **Calculate \( a ** (b ** c) \)**:
- First, compute \( b ** c \):
\[ b ** c = b + bc \]
- Now, substitute this into \( a ** (b ** c) \):
\[ a ** (b ** c) = a ** (b + bc) = a + a(b + bc) = a + ab + abc \]

3. **Compare the two results**:
- From \( (a ** b) ** c \), we have:
\[ a + ab + ac + abc \]
- From \( a ** (b ** c) \), we have:
\[ a + ab + abc \]

Since \( (a ** b) ** c \) includes an additional term \( ac \) that is not present in \( a ** (b ** c) \), we conclude:
\[ (a ** b) ** c \neq a ** (b ** c) \]

Thus, the operation `**` is not associative.

### Final Conclusion
- The operation `**` is a binary operation.
- The operation `**` is not associative.