Show that * on Z + defined

To prove: * is not a binary operation

Given: a and b are defined on positive integer set

And $a * b=|a-b|$

$\Rightarrow a * b=(a-b)$, when $a>b$

$=b-a$ when $b>a$

$=0$ when $a=b$

But 0 is neither positive nor negative.

So 0 does not belong to $Z^{+}$.

So $a * b=|a-b|$ does not belong to $Z+$ for all values of $a$ and $b$

So $*$ is not a binary operation.

Hence proved