Let * be a binary operation on Q0 (set of non-zero rational numbers)

1.	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» 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 *.

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»

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 *.