= Solution
The <random variables> are <independent random variables> if the <events> determined by one are independent of the <events> determined by the other. For <integer>-valued variables, this is equivalent to the factorization
$$
\boxed{p_{X,Y}(x,y)=p_X(x)p_Y(y)\quad\hbox{for every }x,y\in\mathbb Z.}
$$
Indeed, <independence> gives the factorization by taking singleton <events>. Conversely, summing it over $x\in A$ and $y\in B$, for any <subsets> $A,B\subseteq\mathbb Z$, gives $\mathbb P(X\in A,Y\in B)=\mathbb P(X\in A)\mathbb P(Y\in B)$. Thus the singleton condition implies <independence> of all <events> in question.
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