\begin{definition}[Definition:Set Intersection/Finite Intersection]
Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$.
Then:
:$\ds S = \bigcap_{i \mathop \in \N^*_n} S_i := \set {x: \forall i \in \N^*_n: x \in S_i}$
where $\N^*_n = \set {1, 2, 3, \ldots, n}$.
If it is clear from the context that $i \in \N^*_n$, we can also write $\ds \bigcap_{\N^*_n} S_i$.
\end{definition}