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Binary Operation

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Question

An equation $$ on $Z^{+}$ (the set of all non-negative integers) is defined as $a b = a - b, \forall a, b \in Z^{+}$ . Is $*$ a binary operation on $Z^{+}$ ?

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Solution

Operation $$ is defined as $a b = a - b$ , $\forall a, b \in Z^{+}$
$a, b \in Z^{+}$ But, if $b > a$ then $a - b$ becomes negative.
So, it is not true that $a - b \in Z^{+} \forall a, b \in Z^{+}$
Thus, $a * b$ does not belong to $Z^{+}$
Hence, $*$ is not binary on $Z^{+}$

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