Question -
Answer -
First we have to provecommutativity of *
Let a, b ∈ Q0
a * b = (3ab/5)
= (3ba/5)
= b * a
Therefore, a * b = b *a, for all a, b ∈ Q0
Now we have to proveassociativity of *
Let a, b, c ∈ Q0
a * (b * c) = a *(3bc/5)
= [a (3 bc/5)] /5
= 3 abc/25
(a * b) * c = (3 ab/5)* c
= [(3 ab/5) c]/ 5
= 3 abc /25
Therefore a * (b * c)= (a * b) * c, for all a, b, c ∈Q0
Thus * is associativeon Q0
Now we have to findthe identity element
Let e be the identityelement in Z with respect to * such that
a * e = a = e * a ∀ a ∈ Q0
a * e = a and e * a =a, ∀ a ∈ Q0
3ae/5 = a and 3ea/5 =a, ∀ a ∈ Q0
e = 5/3 ∀ a ∈ Q0 [because a is not equal to0]
Thus, 5/3 is theidentity element in Q0 with respect to *.