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⁺, defined * by a * b = a − b

Here, Z⁺ denotes the set of all non-negative integers.

Answer 1

 If a = 1 and b = 2 in Z⁺, then

\[a * b = a - b\] 
\[ = 1 - 2\] 
\[ = - 1 \not\in Z^+ \left[ \because  Z^+ \text{ is the set of non-negative integers } \right]\] 
\[ \Rightarrow \text{Fora} = 1 \text{ and }b = 2, \] 
\[a * b \not\in Z^+ \] 
\[\text{Thus}, * \text{ is not a binary operation on } Z^+ .\]