Didier Piau has already provided advice about how you could improve your post, please do as suggested.
About your question: You need to look up the definition of discrete random variable.
First of all, a (real) random variable needs to be a function from the probability space to (if it is a real one) $\mathbb{R}$. It is possible to use other codomains, you'd need to look up what you are supposed to use.
Second, since the probability space is a discrete space, all random variables are discrete random variables, so we can ignore the "discrete" here.
Third, a random variable needs to be measurable. In the case at hand, it is possible to check this for every provided example by hand. You need to check if $f^{-1} (A)$ for any measurable set $A$ in the codomain is an element of $F$.
Hint: What is $ Z^{-1} (\{1, 4,\})$ ?