Let `'**'` be the binary operation defined on the set Z of all integers as `a ** b = a + b + 1` for all a, b `in Z`. The identity element w.r.t. this operations is

Options

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

Answer

To find the identity element with respect to the binary operation defined as \( a ** b = a + b + 1 \) for all integers \( a \) and \( b \), we will denote the identity element by \( e \). The identity element must satisfy the condition that for any integer \( a \):

1. \( a ** e = a \)
2. \( e ** a = a \)

Let's solve for \( e \):

### Step 1: Set up the equation for the identity element
We start with the first condition:
\[ a ** e = a \]
Substituting the definition of the operation, we have:
\[ a + e + 1 = a \]

### Step 2: Simplify the equation
Now, we will simplify the equation:
\[ a + e + 1 = a \]
Subtract \( a \) from both sides:
\[ e + 1 = 0 \]

### Step 3: Solve for \( e \)
Now, we isolate \( e \):
\[ e = -1 \]

### Step 4: Verify the identity element
To confirm that \( e = -1 \) is indeed the identity element, we check the second condition:
\[ e ** a = a \]
Substituting \( e = -1 \):
\[ -1 ** a = -1 + a + 1 \]
This simplifies to:
\[ a = a \]
This holds true for all integers \( a \).

### Conclusion
Thus, the identity element with respect to the operation \( a ** b = a + b + 1 \) is:
\[ \boxed{-1} \]