Find the identity and invertible elements in Z of the binary operation * defined as a * b = a + b-4, `AA a, b in Z`.

Answer

To solve the problem of finding the identity and invertible elements in \( \mathbb{Z} \) for the binary operation defined as \( a * b = a + b - 4 \), we can follow these steps:

### Step 1: Find the Identity Element
To find the identity element \( e \), we need to satisfy the condition:
\[ a * e = a \]
Substituting the operation into this equation gives:
\[ a + e - 4 = a \]
Now, we can simplify this equation:
\[ e - 4 = 0 \]
From this, we can solve for \( e \):
\[ e = 4 \]
Thus, the identity element is \( e = 4 \).

### Step 2: Find the Inverse Element
Next, we need to find the inverse element for any \( a \in \mathbb{Z} \). The inverse element \( x \) must satisfy:
\[ a * x = e \]
Since we found \( e = 4 \), we can write:
\[ a * x = 4 \]
Substituting the operation gives:
\[ a + x - 4 = 4 \]
Now, we can simplify this equation:
\[ a + x = 4 + 4 \]
This simplifies to:
\[ a + x = 8 \]
Now, we can solve for \( x \):
\[ x = 8 - a \]
Thus, the inverse element of \( a \) is \( x = 8 - a \).

### Summary of Results
- The identity element in \( \mathbb{Z} \) for the operation \( * \) is \( 4 \).
- The inverse element of any \( a \in \mathbb{Z} \) is \( 8 - a \).