Check the commutativity and associativity of each of the following binary operations:
‘*’ on Q defined by a*b = ab2 for all a,b∈Q
Given that * is a binary operation on Q defined by a*b = ab2 for all a,b∈Q.
We know that commutative property is p*q = q*p, where * is a binary operation.
Let’s check the commutativity of given binary operation:
⇒ a*b = ab2
⇒ b*a = ba2
⇒ b*a≠a*b
∴ Commutative property doesn’t holds for given binary operation ‘*’ on ‘Q’.
We know that associative property is (p*q)*r = p*(q*r)
Let’s check the associativity of given binary operation:
⇒ (a*b)*c = (ab2)*c
⇒ (a*b)*c = ab2c2 ...... (1)
⇒ a*(b*c) = a*(bc2)
⇒
⇒ a*(b*c) = ab2c4 ...... (2)
From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘Q’.