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Check commutativity and associativity
Given:
\( a * b = (a – b)^2, \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a * b = (a – b)^2 \)
\( b * a = (b – a)^2 = (-(a – b))^2 = (a – b)^2 \)
✔ Operation is commutative
Associativity:
LHS:
\( (a*b)*c = ((a – b)^2 – c)^2 \)
RHS:
\( a*(b*c) = (a – (b – c)^2)^2 \)
Clearly:
\( ((a – b)^2 – c)^2 \neq (a – (b – c)^2)^2 \)
❌ Operation is NOT associative
Conclusion:
✔ Commutative but ❌ Not associative