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Check commutativity and associativity
Given:
\( a * b = ab + 1, \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a * b = ab + 1 \)
\( b * a = ba + 1 = ab + 1 \)
✔ Operation is commutative
Associativity:
LHS:
\( (a*b)*c = (ab + 1)*c = (ab + 1)c + 1 = abc + c + 1 \)
RHS:
\( a*(b*c) = a*(bc + 1) = a(bc + 1) + 1 = abc + a + 1 \)
Clearly:
\( abc + c + 1 \neq abc + a + 1 \)
❌ Operation is NOT associative
Conclusion:
✔ Commutative but ❌ Not associative