It is given that a * b = ab/4
(i) For a, b, c ∈ Q0
We know that
a * b = ab/4 = ba/4 = b * a
(a * b) * c = ab/ 4 * c = [ab/4 * c]/ 4 = (ab) c/ 16
a * (b * c) = a * bc/4 = [a(bc/4)]/ 4 = a (bc)/16
Here, (ab) c = a (bc)
Therefore, (a * b) * c = a * (b * c)
(ii) Consider e as the identity element and a ∈ Q0
Here, a * e = a
So we get
ae/4 = a where e = 4
Hence, 4 is the identity element in Q.
(iii) Consider a ∈ Q0 which is inverse b
a * b = e
So we get
ab/4 = 4
Here, b = 16/a ∈ Q0
Hence, a ∈ Q0 has 16/a as inverse.