Suppose we have an undirected graph $(V, E)$ and we want to know whether or not it has a vertex cover with at most $s$ vertices.

Let $n = |E|$, and for each edge $e_i = \{u, v\} \in E$ let the $i$th list be $\{\{u\}, \{v\}\}$. Then there is a vertex cover of size at most $s$ if and only if there is a choice of one (singleton) set from each list such that the union of those (singleton) sets has size at most $s$ ─ because these unions are exactly the vertex covers of the original graph.

Suppose we have an undirected graph $(V, E)$ and we want to know whether or not it has a vertex cover with at most $s$ vertices.

Let $m = |V|$ and $n = |E|$, and for each edge $e_i = \{u, v\} \in E$ let the $i$th list be $\{\{u\}, \{v\}\}$. Then there is a vertex cover of size at most $s$ if and only if there is a choice of one (singleton) set from each list such that the union of those (singleton) sets has size at most $s$ ─ because these unions are exactly the vertex covers of the original graph.