The identity element for the binary operation `**` defined on Q - {0} as `a ** b=(ab)/(2), AA a, b in Q - {0}` is

Options

A. 1
B. 0
C. 2
D. None of these

Explanation

To find the identity element for the binary operation defined on \( \mathbb{Q} - \{0\} \) as \( a ** b = \frac{ab}{2} \), we need to determine an element \( e \) such that for every element \( a \) in \( \mathbb{Q} - \{0\} \), the following holds:

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

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

### Step 1: Set up the equation for the identity element
We start with the operation defined as:
\[ a ** e = \frac{ae}{2} \]
We want this to equal \( a \):
\[ \frac{ae}{2} = a \]

### Step 2: Solve for \( e \)
To solve for \( e \), we can multiply both sides of the equation by 2 to eliminate the fraction:
\[ ae = 2a \]
Now, we can divide both sides by \( a \) (since \( a \neq 0 \)):
\[ e = 2 \]

### Step 3: Verify the identity element
Now, we need to verify that \( e = 2 \) works as the identity element. We will check both conditions:

1. **Check \( a ** 2 \)**:
\[ a ** 2 = \frac{a \cdot 2}{2} = a \]
This confirms that \( a ** 2 = a \).

2. **Check \( 2 ** a \)**:
\[ 2 ** a = \frac{2 \cdot a}{2} = a \]
This confirms that \( 2 ** a = a \).

Since both conditions are satisfied, we conclude that the identity element for the operation \( ** \) is indeed \( 2 \).

### Final Answer:
The identity element for the binary operation \( ** \) defined on \( \mathbb{Q} - \{0\} \) is \( 2 \).

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