Definition:Set Intersection/Set of Sets

Definition
Let $\Bbb S$ be a set of sets

The intersection of $\Bbb S$ is:
 * $\displaystyle \bigcap \Bbb S := \set {x: \forall S \in \Bbb S: x \in S}$

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

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

Also denoted as
Some sources denote $\displaystyle \bigcap \mathbb S$ as $\displaystyle \bigcap_{S \mathop \in \mathbb S} S$.

Also see

 * Definition:Union of Set of Sets


 * Intersection of Doubleton for a proof that $\displaystyle \bigcap \set {S, T} = S \cap T$