Collections#

Definition: Collection

A collection / set system / family of sets is a set whose elements are only set.

Definition: Subcollection

A subcollection of a collection \(\mathcal{C}\) is just a subset of \(\mathcal{C}\).

Operations#

Definition: Union of a Collection

The union of a collection \(\mathcal{C}\) is the set of all elements which belong to at least one of the sets in \(\mathcal{C}\).

\[ \{x \mid \exists E\in \mathcal{C} : x\in E \} \]

Notation

Usually, the union of a collection is denoted by

\[ \bigcup \mathcal{C} \]

However, an alternative notation is more useful when we need to consider specific elements of \(\mathcal{C}\). In this case, indexing notation using an index set \(I\) for \(\mathcal{C}\) is used.

\[ \bigcup_{i \in I} \mathcal{C}_i \]

Definition: Intersection of a Collection

The intersection of a collection \(\mathcal{C}\) is the set of elements which belong simultaneously to every set of \(\mathcal{C}\).

\[ \{x \mid \forall E \in \mathcal{C}: x \in \mathcal{C}\} \]

Notation

Usually, the intersection of a collection is denoted by

\[ \bigcap \mathcal{C} \]

However, an alternative notation is more useful when we need to consider specific elements of \(\mathcal{C}\). In this case, indexing notation using an index set \(I\) for \(\mathcal{C}\) is used.

\[ \bigcap_{i \in I} \mathcal{C}_i \]