Question

For each binary operation ∗ defined below, determine whether ∗ is commutative or associative.

$On{Z}^{+},$ define $a\ast b=2^{ab}$

Answer 1

$\text{Checking for binary.}$

Given: $On{Z}^{+},a\ast b=2^{ab}$

$\forall a,b\in Z^{+},2^{ab}\in Z^{+}$

So, $\ast$ is binary.

$\text{Checking for commutative.}$

Given: $On{Z}^{+},a\ast b=2^{ab}$

* is commutative if

$a\ast b=b\ast a$

Now,

$a\ast b=2^{ab}$

And, $b\ast a=2^{ba}$

$=2^{ab}$

$\because a\ast b=b\ast a\forall a,b,\in Z^{+},$

$\ast$ is commutative.

$\text{Checking for associative.}$

$\ast$ is associative if

$(a\ast b)\ast c=a\ast (b\ast c)$

Now, $(a\ast b)\ast c=\left(2^{ab}\right)\ast c$

$=2^{2^{ab}.c}$

And,

$a\ast (b\ast c)=a\ast \left(2^{bc}\right)$

$=2^{a{.2}^{bc}}$

Since $(a\ast b)\ast c\ne a\ast (b\ast c).$

$\ast$ is not an associative binary operation.