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Check commutativity and associativity
Given:
\( a * b = a + ab = a(1 + b), \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a * b = a + ab \)
\( b * a = b + ba \)
Clearly:
\( a + ab \neq b + ab \quad (\text{in general}) \)
❌ Operation is NOT commutative
Associativity:
LHS:
\( (a*b)*c = [a(1+b)] * c = a(1+b)(1+c) \)
RHS:
\( a*(b*c) = a * [b(1+c)] = a[1 + b(1+c)] = a(1 + b + bc) \)
Now simplify LHS:
\( a(1+b)(1+c) = a(1 + b + c + bc) \)
Compare:
\( a(1 + b + c + bc) \neq a(1 + b + bc) \)
❌ Operation is NOT associative
Conclusion:
❌ Neither commutative nor associative