For the binary operation * on `Z` defined by a*b= `a+b+1` the identity element is

Options

A. -2
B. 0
C. 1
D. -1

Explanation

To find the identity element for the binary operation defined on the set of integers \( \mathbb{Z} \) by \( a * b = a + b + 1 \), we will denote the identity element as \( e \). The identity element \( e \) must satisfy the condition that for any integer \( a \):

\[ a * e = a \]
and
\[ e * a = a \]

### Step 1: Set up the equation for the identity element
Using the definition of the operation, we can express the first condition:

\[ a * e = a + e + 1 \]

### Step 2: Set the equation equal to \( a \)
Since \( a * e \) must equal \( a \), we can set up the equation:

\[ a + e + 1 = a \]

### Step 3: Simplify the equation
To isolate \( e \), we can subtract \( a \) from both sides:

\[ e + 1 = 0 \]

### Step 4: Solve for \( e \)
Now, we can solve for \( e \) by subtracting 1 from both sides:

\[ e = -1 \]

### Conclusion
Thus, the identity element for the binary operation \( * \) defined by \( a * b = a + b + 1 \) is:

\[ \boxed{-1} \]