Definition:Set Intersection/Family of Sets

Definition
Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

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


 * $\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

In the context of the Universal Set
In treatments of set theory in which the concept of the universal set is recognised, this can be expressed as follows.

Subsets of General Set
This definition is the same when the universal set $\mathbb U$ is replaced by any set $X$, which may or may not be a universal set:

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.

Also see

 * Definition:Set Union/Family of Sets