Advanced Mathematics

2.2Families of Sets

Purpose of Section

We now extend the operations of union and intersection of sets from two sets to several sets, even an infinite number. In many areas of mathematics, such as topology, analysis, measure theory, probability, and others, finding the intersection and union of large families of sets is common.

2.2.1 Introduction

The union and intersection of sets can be extended to the union and intersection of many sets, even an infinite number. When dealing with a collection of several sets, it is usual practice to refer to them as families or classes of sets. To denote a family of sets, one often uses indices such as {A₁, A₂, A₃, … , A₁₀} and for an infinite family, we might write {A₁, A₂, A₃, …} or

Other common ways to denote families of sets are

where the set Λ is called an index set.

For example

The reader might recall the notation for infinite sums and products as

which motivates the following notation and definition for sets.

Definition Unions and Intersections of Families of Sets

The union of the family of n subsets A₁, A₂, … , A_n in a universe ...