The identity element of a*b = a+b+7 on Z is:

Options

A. 0
B. 7
C. `-7`
D. does not exists.

Answer

To find the identity element of the binary operation defined by \( a * b = a + b + 7 \) on the set of integers \( \mathbb{Z} \), we will follow these steps:

### Step 1: Understand the Definition of Identity Element
An identity element \( E \) for a binary operation \( * \) must satisfy the condition:
\[ a * E = a \quad \text{and} \quad E * a = a \]
for all elements \( a \) in the set.

### Step 2: Apply the Definition to the Given Operation
For our operation, we can write:
\[ a * E = a + E + 7 \]
Setting this equal to \( a \) (as per the definition of the identity element), we have:
\[ a + E + 7 = a \]

### Step 3: Simplify the Equation
To isolate \( E \), we can subtract \( a \) from both sides:
\[ E + 7 = 0 \]

### Step 4: Solve for \( E \)
Now, we can solve for \( E \):
\[ E = -7 \]

### Step 5: Verify the Identity Element
We need to check if \( E = -7 \) satisfies the identity condition:
1. Check \( a * (-7) \):
\[ a * (-7) = a + (-7) + 7 = a \]
2. Check \( (-7) * a \):
\[ (-7) * a = (-7) + a + 7 = a \]

Both conditions hold true, confirming that \( E = -7 \) is indeed the identity element.

### Conclusion
The identity element of the operation \( a * b = a + b + 7 \) on \( \mathbb{Z} \) is \( -7 \).