Show that $*$ on $Z^{+}$defined by $a * b=|a-b|$ is not a binary operation.

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Show that $*$ on $Z^{+}$ defined by $a * b=|a-b|$ is not a binary operation.

Show that $*$ on $Z^{+}$ defined by $a * b=|a-b|$ is not a binary operation.

To prove: $*$ is not a binary operation

Since, a and $b$ are defined on positive integer set

And $\mathrm{a}^{*} \mathrm{~b}=|\mathrm{a}-\mathrm{b}|$

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

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

$=0$ when $\mathrm{a}=\mathrm{b}$

But 0 is neither positive nor negative.

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

So $a * b=\mid a$ – bl does not belong to $Z$ + for all values of a and $b$ ,

So * is not a binary operation.

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