Let R be the set of real numbers other than -1, and * be the binary operation on R defined𝑏𝑦 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏. Determine the identity element and inverse of 𝑎

The identity element of a set is an element that, when combined with any other element in the set under a given operation, leaves the other element unchanged.

For the operation * defined as a * b = a + b + ab, we want to find an element e in R such that for every a in R, a * e = a.

Substituting e into the operation gives us a * e = a + e + ae.

For this to be equal to a, both e and ae must be zero. Since a is any real number other than -1, the only way for ae to be zero is if e is zero.

Substituting e = 0 into the equation gives us a * 0 = a + 0 + a*0 = a, which is what we wanted.

So, the identity element for this operation is 0.

Now, let's find the inverse of a. The inverse of a under the operation * is an element b in R such that a * b = e, where e is the identity element.

Substituting a, b, and e = 0 into the operation gives us a * b = a + b + ab = 0.

Rearranging the equation to solve for b gives us b = -a / (1 + a).

So, the inverse of a under the operation * is -a / (1 + a).

The identity element of a set with a binary operation is an element that, when applied to any element in the set, leaves it unchanged.

Let's denote the identity element as 'e'. According to the given operation, we have:

a * e = a + e + ae = a

This simplifies to:

a + e + ae = a ae + e = 0 e(a + 1) = 0

Since a is any real number other than -1, a + 1 ≠ 0. Therefore, the only way for the equation to hold is if e = 0. So, the identity element is 0.

Now, let's find the inverse of an element 'a'. The inverse of 'a' is an element 'b' such that a * b = e, where 'e' is the identity element.

According to the given operation, we have:

a * b = a + b + ab = 0

This simplifies to:

a + b + ab = 0 b(a + 1) = -a b = -a / (a + 1)

So, the inverse of 'a' is -a / (a + 1).