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

Answer 1

To find the inverse of an element \( a \) under the binary operation defined by \( a * b = a + b + ab \), we first need to verify the properties of this operation. The operation has been stated to have a neutral element of 0. This means that for any number \( a \):

\[
a * 0 = a + 0 + a \cdot 0 = a
\]

Now, we need to find the inverse of an element \( a \). An inverse \( b \) of an element \( a \) under the operation \( * \) must satisfy:

\[
a * b = 0
\]

We can substitute the definition of our operation into this equation:

\[
a * b = a + b + ab = 0
\]

Rearranging gives:

\[
b + ab = -a
\]

Factoring out \( b \) from the left side:

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

Now, solving for \( b \):

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

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

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

This is valid as long as \( 1 + a \neq 0 \) (which occurs when \( a \neq -1 \)). When \( a = -1\), the operation does not define an inverse as it results in a division by zero. Therefore, the inverse of \( a \) under the defined operation is:

\[
\frac{-a}{1 + a} \text{ for } a \neq -1.
\]