Definition:Set Intersection/Family of Sets/Universal Set

Definition
Let $\mathbb U$ be a universal set.

Let $I$ be an indexing set.

Let $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be a family of subsets of $\mathbb U$.

Then the intersection of $\left \langle {S_i} \right \rangle$ is defined as:


 * $\displaystyle \bigcap_{i \mathop \in I} S_i := \left\{{x \in \mathbb U: \forall i \in I: x \in S_i}\right\}$

Thus $S \cap T$ can be defined as:
 * $S \cap T := \left\{{x \in \mathbb U: x \in S \land x \in T}\right\}$

Also denoted as
The set $\displaystyle \bigcap_{i \mathop \in I} S_i$ can also be seen denoted as:


 * $\displaystyle \bigcap_I S_i$

or, if the indexing set is clear from context:


 * $\displaystyle \bigcap_i S_i$

However, on this website it is recommended that the full form is used.