Definition:Set Intersection/Family of Sets/Subsets of General Set

Definition

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $X$.

Then the intersection of $\family {S_i}$ is defined as:

$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$

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:

$\displaystyle \bigcap_i S_i$

The form:

$\displaystyle \bigcap_{S \mathop \in X} S$

can also be seen, but this obscures the true nature of the indexing set.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ it is recommended that the full form is used.