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