Definition:Set Intersection/Family of Sets/Universal Set

Definition

Let $\mathbb U$ be a universal set.

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$.

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

$\ds \bigcap_{i \mathop \in I} S_i := \set {x \in \mathbb U: \forall i \in I: x \in S_i}$

Also denoted as

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

$\ds \bigcap_I S_i$

or, if the indexing set is clear from context:

$\ds \bigcap_i S_i$

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

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets