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