If * is a operation on A = {- 1, 1} such that a * b = ab ∀ a, b ∈ A then find the identity element of A with respect to operation * ?

Concept:

Let * be a binary operation on a non-empty set S. If there exists an element e in S such that a * e = e * a = a ∀ a ∈ S. Then the element e is said to be an identity element of S with respect to *.

Calculation:

Given: A = {1, - 1} and * is a operation on A such that a * b = ab ∀ a, b ∈ A.

Let e be the identity element of A with respect to *.

As we know that if e is an identity element of a non-empty set S with respect to a binary operation * then a * e = e * a = a ∀ a ∈ S.

Let a ∈ A and because e is the identity element of A with respect to given operation *.

.e a * e = a =  e * a ∀ a ∈ A.

According to the definition of *, we have

⇒ a * e = ae = a

⇒ e = 1 ∈ A

Hence, 1 is the identity element of A with respect to given operation *.