Consider a binary operation on Q – {1}, defined by a * b = a + b - ab.

(i) Find the identity element in Q – {1}.

(ii) Show that each a ∈ Q - {1} has its inverse.

(i) For a binary operation *, e identity element exists if a*e = e*a = a. As a*b = a+ b- ab

a*e = a+ e- ae (1)

e*a = e+ a- e a (2)

using a*e = a

a+ e- ae = a

e-ae = 0

e(1-a) = 0

either e = 0 or a = 1 as operation is on Q excluding 1 so a≠1, hence e = 0.

So identity element e = 0.

(ii) for a binary operation * if e is identity element then it is invertible with respect to * if for an element b, a*b = e = b*a where b is called inverse of * and denoted by a^-1.

a*b = 0

a+ b- ab = 0

b(1-a) = -a