Let 'x' be an operation defined as `x:RxxRrarrR` Such that `a**b=2a+b,a,binR`
Check if `'**'` is a binary operation
If yes, find if it is associative too.

Answer

To determine if the operation \( ** \) defined by \( a ** b = 2a + b \) 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 \( S \) is a function that combines two elements of \( S \) to produce another element of \( S \). In this case, our set is the real numbers \( \mathbb{R} \).

1. **Take two arbitrary elements \( a \) and \( b \) from \( \mathbb{R} \)**.
2. **Apply the operation**:
\[ a ** b = 2a + b \]
3. **Check if the result is also in \( \mathbb{R} \)**:
- Since \( a \) and \( b \) are real numbers, \( 2a \) is also a real number (as the product of a real number and a constant is a real number).
- The sum \( 2a + b \) is also a real number (as the sum of two real numbers is a real number).

Thus, since \( a ** b \) results in a real number for any real numbers \( a \) and \( b \), we conclude that \( ** \) is a binary operation.

### Step 2: Check if \( ** \) is associative

An operation is associative if for all \( a, b, c \in \mathbb{R} \):
\[ (a ** b) ** c = a ** (b ** c) \]

1. **Calculate \( (a ** b) ** c \)**:
- First, find \( a ** b \):
\[ a ** b = 2a + b \]
- Now apply the operation with \( c \):
\[ (a ** b) ** c = (2a + b) ** c = 2(2a + b) + c = 4a + 2b + c \]

2. **Calculate \( a ** (b ** c) \)**:
- First, find \( b ** c \):
\[ b ** c = 2b + c \]
- Now apply the operation with \( a \):
\[ a ** (b ** c) = a ** (2b + c) = 2a + (2b + c) = 2a + 2b + c \]

3. **Compare the two results**:
- From \( (a ** b) ** c \), we have:
\[ 4a + 2b + c \]
- From \( a ** (b ** c) \), we have:
\[ 2a + 2b + c \]

Since \( 4a + 2b + c \neq 2a + 2b + c \) for general \( a \), we conclude that:
\[ (a ** b) ** c \neq a ** (b ** c) \]

Thus, the operation \( ** \) is not associative.

### Final Conclusion

- The operation \( ** \) is a binary operation.
- The operation \( ** \) is not associative.