The problem asks which of the given binary operations is commutative. A binary operation $@$ is commutative if $a @ b = b @ a$ for all $a$ and $b$. We need to test each of the given options to see if this condition holds.

1. Problem Description

The problem asks which of the given binary operations is commutative. A binary operation $@$ is commutative if $a @ b = b @ a$ for all $a$ and $b$. We need to test each of the given options to see if this condition holds.

2. Solution Steps

A. $a @ b = a + b - ab$

$b @ a = b + a - ba = a + b - ab$

Since $a @ b = b @ a$, this operation is commutative.

B. $a @ b = 2a + 2b - ab$

$b @ a = 2b + 2a - ba = 2a + 2b - ab$

Since $a @ b = b @ a$, this operation is commutative.

C. $a @ b = \frac{1}{a} + \frac{1}{b}$

$b @ a = \frac{1}{b} + \frac{1}{a} = \frac{1}{a} + \frac{1}{b}$

Since $a @ b = b @ a$, this operation is commutative.

D. $a @ b = a - b + ab$

$b @ a = b - a + ba = b - a + ab$

In general, $a - b + ab \ne b - a + ab$. For example, let $a = 1$ and $b = 2$. Then $a @ b = 1 - 2 + (1)(2) = 1 - 2 + 2 = 1$, while $b @ a = 2 - 1 + (2)(1) = 2 - 1 + 2 = 3$. Since $a @ b \ne b @ a$, this operation is not commutative.

Since options A, B, and C are commutative, there might be an error in the problem statement. Assuming the problem asks for the only commutative option, we re-evaluate each answer.

A. $a @ b = a + b - ab$

$b @ a = b + a - ba = a + b - ab$

Thus, $a @ b = b @ a$, and the operation is commutative.

B. $a @ b = 2a + 2b - ab$

$b @ a = 2b + 2a - ba = 2a + 2b - ab$

Thus, $a @ b = b @ a$, and the operation is commutative.

C. $a @ b = \frac{1}{a} + \frac{1}{b}$

$b @ a = \frac{1}{b} + \frac{1}{a}$

Thus, $a @ b = b @ a$, and the operation is commutative.

D. $a @ b = a - b + ab$

$b @ a = b - a + ba$

Thus, we need to check $a - b + ab = b - a + ab$. This is equivalent to $a - b = b - a$, which means $2a = 2b$, or $a = b$. This is not true for all $a$ and $b$. Therefore, the operation is not commutative.

Options A, B and C are all commutative, so the intended answer is likely that only one is supposed to be commutative. However, given the options, A, B and C are commutative, and D is not. There could be a typo in the definition of either A, B, or C.

Let's assume there is a typo and consider an additional case for each of A, B, and C:

A. If $a @ b = a - b - ab$, then $b @ a = b - a - ba$. Since $a - b - ab \ne b - a - ab$, this operation is not commutative.

B. If $a @ b = 2a - 2b - ab$, then $b @ a = 2b - 2a - ba$. Since $2a - 2b - ab \ne 2b - 2a - ab$, this operation is not commutative.

C. If $a @ b = \frac{1}{a} - \frac{1}{b}$, then $b @ a = \frac{1}{b} - \frac{1}{a}$. Since $\frac{1}{a} - \frac{1}{b} \ne \frac{1}{b} - \frac{1}{a}$, this operation is not commutative.

If we assume the question asks "Which of the following binary operations is NOT commutative?", then the answer would be D. However, if we assume the question is indeed asking "Which of the following binary operations IS commutative?", then A, B and C would all be correct. Based on the choices, I believe D is intended to be the ONLY non-commutative operation.

3. Final Answer

D