If `** ` on Q defined by `a**b = a - b ` . Identity element is

Options

A. 0
B. 1
C. a - b
D. Does not exists

Answer

To find the identity element for the operation defined on \( Q \) by \( a ** b = a - b \), we need to determine an element \( e \) such that for any element \( a \) in \( Q \):

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

Let's go through the steps to find the identity element.

### Step 1: Define the operation
The operation is defined as:
\[ a ** b = a - b \]

### Step 2: Find the left identity
We need to find \( e \) such that:
\[ a ** e = a \]
Substituting the operation:
\[ a - e = a \]
To isolate \( e \), we can rearrange the equation:
\[ -e = a - a \]
\[ -e = 0 \]
Thus, we find:
\[ e = 0 \]

### Step 3: Verify the left identity
Now we need to check if \( e = 0 \) is indeed a left identity:
\[ a ** 0 = a - 0 = a \]
This holds true for any \( a \) in \( Q \).

### Step 4: Find the right identity
Next, we check if \( e \) is also a right identity:
\[ e ** a = a \]
Substituting the operation:
\[ e - a = a \]
Rearranging gives:
\[ e = a + a \]
\[ e = 2a \]
This means that for different values of \( a \), \( e \) changes, which implies that \( e \) cannot be a single constant value for all \( a \).

### Step 5: Conclusion
Since we found that \( e = 0 \) works as a left identity but does not work as a right identity (as it depends on \( a \)), we conclude that there is no identity element for the operation defined by \( a ** b = a - b \).

### Final Answer
The identity element does not exist for the operation \( a ** b = a - b \).

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