Question

Let * be a binary operation on Q₀ (set of non-zero rational numbers) defined by a * b = (3ab/5) for all a, b ∈ Q₀. Show that * is commutative as well as associative. Also, find its identity element, if it exists.

Answer 1

Let us prove the commutativity of *

Let a, b ∈ Q₀

a * b = (3ab/5)

= (3ba/5)

= b * a

So, a * b = b * a, for all a, b ∈ Q₀

Let us prove the associativity of *

Let a, b, c ∈ Q₀

a * (b * c) = a * (3bc/5)

= [a(3bc/5)]/5

= 3 abc/25

(a * b) * c = (3ab/5) * c

= [(3ab/5)c]/5

= 3abc/25

So, a * (b * c) = (a * b) * c, for all a, b, c ∈ Q₀

Thus * is associative on Q₀

Let us find the identity element

Let e be the identity element 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 ! = 0]

Hence, 5/3 is the identity element in Q₀ with respect to *.