A binary operation is commutative on a set if the order of the operands does not affect the result. In other words, for any two elements a and b in the set, the operation a * b must be equal to b * a.

Let’s check each of the given binary operations to see if it is commutative on the set of integers:

A. The operation a * b = a + 2b is not commutative, because in general a + 2b is not equal to b + 2a. For example, if a = 1 and b = 2, then a * b = 1 + 2 * 2 = 5, but b * a = 2 + 2 * 1 = 4.

B. The operation a * b = a + b - ab is commutative, because a + b - ab is equal to b + a - ba. This can be seen by simply rearranging the terms: a + b - ab = b + a - ba.

C. The operation a * b = a^2 + b is not commutative, because in general a^2 + b is not equal to b^2 + a. For example, if a = 1 and b = 2, then a * b = 1^2 + 2 = 3, but b * a = 2^2 + 1 = 5.

D. The operation a * b = (a(b + 1))/2 is not commutative, because in general (a(b + 1))/2 is not equal to (b(a + 1))/2. For example, if a = 1 and b = 2, then a * b = (1(2 + 1))/2 = 3/2, but b * a = (2(1 + 1))/2 = 4/2 = 2.

So, the binary operation a (\ast) b = a + b - ab is commutative on the set of integers.