Here is the problem. Given $k, n, T\_1, \ldots, T\_m$, where each $T\_i \subseteq \{1, \ldots, n\}$. Is there a subset $S \subseteq \{1, \ldots, n\}$ with size at most $k$ such that $S \cap T\_i \neq \emptyset$ for all $i$? I am trying to reduce this problem to SAT. My idea of a solution would be to have a variable $x\_i$ for each of 1 to $n$. For each $T\_i$, create a clause $(x\_{i\_1} \vee \cdots \vee x\_{i\_k})$ if $T\_i = \{i\_1, \ldots, i\_k\}$. Then and all these clauses together. But this clearly isn't a complete solution as it does not represent the constraint that $S$ must have at most $k$ elements. I know that I must create more variables, but I'm simply not sure how. So I have two questions:

1. Is my idea of solution on the right track?
2. How should the new variables be created so that they can be used to represent the cardinality $k$ constraint?