Indexed Family of Sets

If we have a collection of sets represented as a single set, we call it a family of sets. Usually, the sets in a family share some common context, though it is unnecessary. For example, if we have a set $S$ and another set $\cal F$ containing some subsets of the set $S$, we call $\cal F$ a family of sets.

What is an indexed family of sets?

We may index the members of a family of sets. An indexed family of sets $\cal A$ is a collection of sets associated with a specific set $I$ called an indexing set. There is a bijection between the elements in $I$ and the sets in $\cal A$. So, each element in $I$ corresponds to exactly one set in $\cal A$. In such a scenario, we say that the family $\cal A$ is indexed by the set $I$.

Formally, a family of sets $\cal A$ indexed by a set $I$ is represented in the set-builder notation as follows: