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.

Here's how to analyze the given operation:

The operation is defined on the set of integers $\mathbb{Z}$ as $x * y = xy - x - y$.

Step 1: Determine if the operation is commutative. 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.

Step 2: Find the identity element. 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.

Step 3: Find the inverse of $x$. 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.

3 done, 2 left today. You're making progress.