Let A be the set of all real numbers except -1 and an operation 'o' be defined on A by aob = a+b + ab for all a, `b in A`, then identify elements w.r.t. 'o' is

Options

A. a
B. b
C. 1
D. 0

Answer

To identify the identity element with respect to the operation 'o' defined on the set A, we follow these steps:

### Step-by-Step Solution:

1. **Understand the Operation**:
The operation 'o' is defined as:
\[ a \, o \, b = a + b + ab \]
for all \( a, b \in A \), where \( A \) is the set of all real numbers except -1.

2. **Identify the Identity Element**:
An identity element \( e \) with respect to the operation 'o' must satisfy the condition:
\[ a \, o \, e = a \]
for all \( a \in A \).

3. **Substituting into the Operation**:
Substitute \( e \) into the operation:
\[ a \, o \, e = a + e + ae \]
We want this to equal \( a \):
\[ a + e + ae = a \]

4. **Simplifying the Equation**:
By subtracting \( a \) from both sides, we get:
\[ e + ae = 0 \]

5. **Factoring the Equation**:
Factor out \( e \):
\[ e(1 + a) = 0 \]

6. **Finding Possible Values for e**:
From this equation, we have two possibilities:
- \( e = 0 \)
- \( 1 + a = 0 \) (which gives \( a = -1 \), but -1 is not in the set A)

7. **Conclusion**:
Since \( a = -1 \) is not a valid option in set A, the only possible identity element is:
\[ e = 0 \]

Thus, the identity element with respect to the operation 'o' is \( 0 \).