Question

For each binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(iv) On $Z^{+}$, define $a\ast b=2^{ab}$

Answer 1

On $Z^{+}$, define $a\ast b=2^{ab}$
$2^{ab}\in Z^{+}$, so the operation $\ast$ is binary operation
It is known that
ab=ba for $a,b\in Z^{+}$
Therefore, $2^{ab}=2^{ba}fora,b\in Z^{+}$
Therefore, $a\ast b=b\ast a$ for $a,b\in Z^{+}$
Therefore, the operation $\ast$ is commutative. It can be observed that
$(1\ast 2)\ast 3=2^{(1\times 2)}\ast 3=4\ast 3=2^{4\times 3}=2^{12}$
$1\ast (2\ast 3)=1\ast 2^{2\times 3}=1\ast 2^6=1\ast 64=2^{64}$
Therefore, $(1\ast 2)\ast 3\ne 1\ast (2\ast 3)$, where $1,2,3,\in Z^{+}$
Therefore, the operation $\ast$ is not associative.