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 1

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