Definition:Set Intersection/Set of Sets

Definition
Let $I$ be an indexing set.

Let $\left \langle {X_i} \right \rangle_{i \mathop \in I}$ be a family of subsets of a set $S$.

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


 * $\displaystyle \bigcap_{i \mathop \in I} X_i = \left\{{y: \forall i \in I: y \in X_i}\right\}$

This notation can also be used as $\displaystyle \bigcap_i X_i$ to be written $\displaystyle \bigcap_{i \mathop \in I} X_i$.

The indexing set itself can be disposed of, as follows:

If $\Bbb S$ is a set of sets, then the intersection of $\Bbb S$ is:
 * $\displaystyle \bigcap \Bbb S = \left\{{x: \forall S \in \Bbb S: x \in S}\right\}$

That is, the set of all objects that are elements of all the elements of $\Bbb S$.

Thus:
 * $\displaystyle S \cap T = \bigcap \left\{{S, T}\right\}$