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 $a * b = a$.

(A) On $Z^{+}$,the operation $*$ is defined by $a * b = a$.
For any two elements $a, b \in Z^{+}$,the result of the operation $a * b$ is $a$.
Since $a$ is an element of $Z^{+}$,it follows that for every pair $(a, b) \in Z^{+} \times Z^{+}$,there exists a unique element $a * b$ in $Z^{+}$.
By definition,a binary operation on a set $S$ is a function from $S \times S$ to $S$.
Since $a * b = a \in Z^{+}$ for all $a, b \in Z^{+}$,the operation $*$ satisfies the condition of being a binary operation.
Therefore,$*$ is a binary operation on $Z^{+}$.