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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.
Solution
\[\ a, b \in Z^+ \]
\[ \Rightarrow \left| a - b \right| \in Z^+ \left[ \because\left| a - b \right|\text{is a} \text{ positive integer} \right]\]
\[ \Rightarrow a * b \in Z^+ \]
\[\text{ Therefore },\]
\[a * b \in Z^+ , \forall a, b \in Z^+ \]
\[\text{Thus}, * \text{ is a binary operation on } Z^+ .\]
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