Determine whether the binary operation defined on Z by lett

Determine whether the binary operation * defined on Z by letting a * b = 2^ab is commutative and whether it is associative. Prove your assertion.

Solution

a*b=2^{ab}

b*a=2^{ba}=2^{ab}=a*b

Hence it is commutative

a*(b*c)=a*2^{bc}=2^{a2^{bc}}

(a*b)*c=2^{ab}*c=2^{2^{ab}c}

Hence no associative