(i) Consider e as an identity element.
We know that a * e = a Ɐ a ∈ Q {1}
So a + e – ae = a
It can be written as
e (1 – a) = 0
So e = 0 ∈ Q {1}
a * 0 = a + 0 = a
So 0 * a = 0 + a = a
Hence, 0 is the identity element in Q – {1}
(ii) Consider a ∈ Q – {1} where a -1 = b
We know that a * b = 0
It can be written as
a + b – ab = 0
So a = ab – b
a = (a – 1) b
We get b = a/ a – 1 ∈ Q – {1}
So a -1 = a/ a – 1 ∈ Q – {1}
Therefore, each a ∈ Q – {1} has its inverse.