Question:
Show that $*$ on $Z+$ defined by $a * b=|a-b|$ is not a binary operation.
Solution:
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