Question

A binary operation* is defined on the set of real numbers by a *b= a +b + ab. Of the neutral element under the operation is 0 find the inverse of the element a€b and identity element

Answer 1

To analyze the binary operation defined as \( a * b = a + b + ab \), let's first confirm whether the operation has an identity element and then find the inverse of any element \( a \).

### Step 1: Identity Element

To find the identity element \( e \) for this operation, we need to satisfy the condition:

\[
a * e = a \quad \text{and} \quad e * a = a
\]

Using the operation definition, we start with \( a * e \):

\[
a * e = a + e + ae
\]

We want this to equal \( a \):

\[
a + e + ae = a
\]

Subtracting \( a \) from both sides gives:

\[
e + ae = 0
\]

Factoring out \( e \):

\[
e(1 + a) = 0
\]

For this equation to hold for all \( a \), the only solution is \( e = 0 \).

Thus, the identity element for this operation is \( 0 \).

### Step 2: Inverse Element

Next, we find the inverse of any element \( a \). The inverse \( b \) of an element \( a \) must satisfy:

\[
a * b = 0
\]

Using our operation definition:

\[
a * b = a + b + ab
\]

Setting it equal to \( 0 \):

\[
a + b + ab = 0
\]

Rearranging gives:

\[
ab + a + b = 0
\]

This can be rewritten as:

\[
ab + a + b + 1 = 1
\]

Factoring, we get:

\[
(a + 1)(b + 1) = 1
\]

From this, we can express \( b + 1 \) in terms of \( a \):

\[
b + 1 = \frac{1}{a + 1}
\]

Thus,

\[
b = \frac{1}{a + 1} - 1
\]

Simplifying gives:

\[
b = \frac{1 - (a + 1)}{a + 1} = \frac{-a}{a + 1}
\]

Thus, the inverse of the element \( a \) under the operation \( * \) is:

\[
b = -\frac{a}{a + 1}
\]

### Summary

1. **Identity Element**: \( 0 \)
2. **Inverse of element \( a \)**: \( -\frac{a}{a + 1} \) (provided \( a \neq -1 \) to avoid division by zero).