The operation is xy=x+y-xy Not xy=x-y-xy

Thanks for the correction! Let's re-evaluate the operation x * y = x + y - xy. Step 1: Determine if the operation is commutative. An operation is commutative if x y = y x for all x, y Z. Given x * y = x + y - xy. Now, let's find y * x: y * x = y + x - yx Since addition and multiplication of integers are commutative (x+y = y+x and xy = yx), we can see that: x + y - xy = y + x - yx Therefore, x y = y x. The operation is commutative. Step 2: Find the identity element. An identity element e Z for the operation must satisfy x e = x for all x Z. Using the definition of the operation: x + e - xe = x Subtract x from both sides: e - xe = 0 Factor out e: e(1 - x) = 0 For this equation to hold for all x Z, the value of e must be 0. If e = 0, then 0(1 - x) = 0, which simplifies to 0 = 0. This is true for all integers x. Therefore, the identity element is e = 0. Step 3: Find the inverse of x. An inverse element x^-1 for an element x exists if x * x^-1 = e, where e is the identity element. We found e = 0. So, we need to solve x * x^-1 = 0: x + x^-1 - xx^-1 = 0 Factor out x^-1 from the terms containing it: x + x^-1(1 - x) = 0 x^-1(1 - x) = -x If 1 - x ≠ 0 (i.e., x ≠ 1), we can solve for x^-1: x^-1 = (-x)/(1 - x) = (x)/(x - 1) For x^-1 to be an inverse in Z, it must be an integer. Let's test some values: • If x = 0, x^-1 = (0)/(0 - 1) = 0. (This is an integer). • If x = 2, x^-1 = (2)/(2 - 1) = 2. (This is an integer). • If x = 3, x^-1 = (3)/(3 - 1) = (3)/(2). (This is not an integer). • If x = -1, x^-1 = (-1)/(-1 - 1) = (-1)/(-2) = (1)/(2). (This is not an integer). Also, consider the case when x = 1: If x = 1, the equation x + x^-1 - xx^-1 = 0 becomes: 1 + x^-1 - 1 · x^-1 = 0 1 + x^-1 - x^-1 = 0 1 = 0 This is a contradiction, which means that x=1 does not have an inverse. Since x^-1 is not an integer for all x Z (e.g., for x=3) and does not exist for x=1, the inverse of x does not exist for all x Z. Summary of findings: • The operation is commutative. • The identity element is e = 0. • The inverse of x does not exist for all x Z. What's next?