Question
Determine whether or not the definition of * given below gives a binary operation.
In the event that * is not a binary operation give justification of this.
On $Z ^{+}$define * by $a * b = |a - b|$
Here, $Z^{+}$denotes the set of all non-negative integers.

Answer

On $Z ^{+}{ }^*$ is defined by $a ^* b=| a - b |$.
It is seen that for each $a , b \in Z ^{+}$,
there is a unique element $| a - b |$ in $Z ^{+}$.
This means that * carries each pair $(a, b)$ to a unique element $a^* b=|a-b|$ in $Z^{+}$.
Therefore, * is a binary operation.

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