I've come with an idea for reduction from the set-cover problem to the vertex-cover problem, But I'm not sure if this reduction is correct. I saw in this post's comments that "There cannot be a nice reduction of this sort".
Consider the set of $S$ to be $S_1,...,S_n$ where the element of $S$ is $s_1, ..., s_m$. In the graph $G=(V,E)$ for each $S_i$ there is a vertex, and an edge from $S_i$ to $S_j$ will be iff there is some element $x\in S_i \land x\in S_j$.
is this reduction even right?