= Solution
{c}
For each nonnegative integer $z$, the <events> $\{X=z-y,Y=y\}$, $0\le y\le z$, are pairwise disjoint and their union is $\{X+Y=z\}$. By <countable additivity> and then <independence>,
$$
\begin{aligned}
p_Z(z)&=\sum_{y=0}^z\mathbb P(X=z-y,Y=y)\\
&=\boxed{\sum_{y=0}^zp_Y(y)p_X(z-y).}
\end{aligned}
$$
This is the <discrete convolution> of the two <probability mass functions>. If the convention for natural numbers excludes zero, set both <probability mass functions> equal to zero there; the formula remains valid. Also $p_Z(z)=0$ for negative $z$.
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