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 F containing some subsets of the set S, we call 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 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 A. So, each element in I corresponds to exactly one set in A. In such a scenario, we say that the family A is indexed by the set I.
Formally, a family of sets A indexed by a set I is represented in the set-builder notation as follows: