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An Introduction to Basic Set Theory

Indexed Family of Sets

Explore the concept of indexed families of sets, where each set is associated with an element in an indexing set. Understand how to perform union and intersection on these collections, and see how distributive and DeMorgan's laws apply. This lesson helps you grasp how indexing organizes complex sets and how set operations extend to infinite families.

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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 SS and another set F\cal F containing some subsets of the set SS, we call F\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 A\cal A is a collection of sets associated with a specific set II called an indexing set. There is a bijection between the elements in II and the sets in A\cal A. So, each element in II corresponds to exactly one set in A\cal A. In such a scenario, we say that the family A\cal A is indexed by the set II.

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

Here, ...