Check the commutativity and associativity of each of the following binary operations:
‘*’ on Q defined by a*b = (a – b)2 for all a,b∈Q
Given that * is a binary operation on Q defined by a*b = (a – b)2 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 = (a – b)2
⇒ b*a = (b – a)2 = (a – b)2
⇒ b*a = a*b
∴ Commutative property 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 = ((a – b)2)*c
⇒ 
⇒ (a*b)*c = (a2 + b2 – 2ab – c)2 ...... (1)
⇒ a*(b*c) = a*((b – c)2)
⇒ 
⇒ a*(b*c) = (a2 – b2 – c2 + 2bc)2 ...... (2)
From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘Q’.