Check the commutativity and associativity of each of the following binary operations:

‘*’ on Q defined by a*b = ab2 for all a,bQ

Given that * is a binary operation on Q defined by a*b = ab2 for all a,bQ.


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’.


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