Determine whether the binary operation $$ on the set N of natural numbers defined by $a b = 2^{a b}$ is associative or not.

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Solution

$1 * (2 * 3) = 1 * 2^{6} = 1 * 64 = 2^{64}, (1 * 2) * 3 = 2^{2} * 3 = 4 * 3 = 2^{12} .$
Clearly $1 * (2 * 3) \neq (1 * 2) * 3.$ Hence $*$ is not associative.

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