Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = ab$.

(A) On $Z^{+}$,$*$ is defined by $a * b = ab$.
It is observed that for any two elements $a, b \in Z^{+}$,their product $ab$ is also a positive integer,meaning $ab \in Z^{+}$.
Since the product of two positive integers is always a unique positive integer,the operation $*$ maps every pair $(a, b)$ to a unique element $a * b = ab$ in $Z^{+}$.
Therefore,$*$ is a binary operation on $Z^{+}$.