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Check commutativity and associativity
Given:
\( a * b = a^b, \quad a,b \in \mathbb{N} \)
Commutativity:
\( a * b = a^b \)
\( b * a = b^a \)
Example:
\( 2^3 = 8 \neq 3^2 = 9 \)
❌ Operation is NOT commutative
Associativity:
LHS:
\( (a*b)*c = (a^b)^c = a^{bc} \)
RHS:
\( a*(b*c) = a^{(b^c)} \)
Example:
\( (2^3)^2 = 8^2 = 64 \)
\( 2^{(3^2)} = 2^9 = 512 \)
❌ Operation is NOT associative
Conclusion:
❌ Neither commutative nor associative