I want to figure out whether my problem described below is reducible to the set cover problem I have a collection of non-empty sets $S_i$, where $1 \le i \le n$, which are constructed from elements of a universe $U$.

I need an algorithm that gives me a set $T$ of minimum cardinality ($T \subseteq U$), such that $T \cap S_i \ne \phi, \forall i$, where $1 \le i \le n$.

I am not sure whether it is an NP-complete problem and can be reduced to minimum set cover in some way. I thought of reducing it to vertex cover, by taking the sets as vertices, and putting an edge between any two sets if their intersection is non-empty, but that doesn't quite seem to work either.
Hint: For each element $x \in U$, let $A_x = \{i : x \in S_i\}$. Then $T$ is a solution to your problem iff $\{A_x : x \in T\}$ covers all of $\{1,\ldots,n\}$. This problem is known as minimum hitting set.