Question

Let * be a binary operation on Q defined by a*b = 3ab/5. Show that * is commutative as well as associative. Also find its identity element, if it exists.

Answer 1

For commutativity, condition that should be fulfilled is a * b = b * a

Consider a*b = 3ab/5 = 3ba/5 = b*a

∴ a*b = b*a

Hence, * is commutative. 

For associativity, condition is (a * b) * c = a * (b * c)

Consider (a*b)*c = (3ab/5)*c = 9ab/25

and a*(b*c) = a*(3bc/5) = 9ab/25

Hence, (a * b) * c = a * (b * c) 

∴ * is associative. 

Let e ∈ Q be the identity element, 

Then a * e = e * a = a

⇒ 3ae/5 = 3ea/5 = a ⇒ e = 5/3