Let Z be the set of integers and define the operation on Z as xy = xy − x − y + 2. Find (i) the identity element and (ii) the inverse of x in Z.

An operation is commutative if $x * y = y * x$ for all $x, y \in \mathbb{Z}$.
Given $x * y = xy - x - y$.
Now, let's find $y * x$:

$$y * x = yx - y - x$$

Since multiplication of integers is commutative ($xy = yx$) and addition/subtraction is commutative ($-x - y = -y - x$), we can see that:

$$xy - x - y = yx - y - x$$

Therefore, $x * y = y * x$.
The operation is commutative.

An identity element $e \in \mathbb{Z}$ for the operation $*$ must satisfy $x * e = x$ for all $x \in \mathbb{Z}$.
Using the definition of the operation:

$$xe - x - e = x$$

Add $x$ to both sides:

$$xe - e = 2x$$

Factor out $e$:

$$e(x - 1) = 2x$$

For $e$ to be an identity element, it must be a fixed integer value, independent of $x$.
Consider the case when $x = 1$:

$$e(1 - 1) = 2(1)$$ $$e(0) = 2$$ $$0 = 2$$

This is a contradiction. Since the equation $e(x-1) = 2x$ has no solution for $e$ when $x=1$, there is no single identity element that works for all integers in $\mathbb{Z}$.
Therefore, the identity element does not exist for this operation on the set of integers.

An inverse element $x^{-1}$ for an element $x$ exists only if there is an identity element $e$ such that $x * x^{-1} = e$ and $x^{-1} * x = e$.
Since we have determined that there is no identity element $e$ for the operation $*$ on the set of integers $\mathbb{Z}$, the concept of an inverse element is not applicable.
Therefore, the inverse of $x$ does not exist.

Summary of findings:

• The operation is commutative.
• The identity element does not exist.
• The inverse of $x$ does not exist.