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.

$$\bigcap _{i\in I}{\mathcal{C}}_i$$