18Collections of sets

We saw in Section 4 that a set can have other sets among its elements. This section collects a few things worth saying about sets of that kind.

Given a set X, all of the subsets of X can themselves be gathered into a single set. We call it the power set of X, and we write it 𝒫(X). So the inclusion criterion for 𝒫(X) is Y ⊂ X:

Definition 13 (Power set). 𝒫(X) = {Y : Y ⊂ X}.

For example, 𝒫({7, 8}) = {∅, {7}, {8}, {7, 8}}. Notice that ∅ and X are elements of 𝒫(X) whatever X is, by Propositions 4 (an empty set is a subset of anything) and 2 (every set is a subset of itself).

We often want to take the union or the intersection of more than two sets at once. Let ℱ be a collection of sets, by which I just mean a set whose elements are themselves sets:

Definition 14 (Collection). A collection is a set whose elements are themselves sets.

The union of ℱ is the set of objects belonging to at least one element of ℱ, and we write it ⋃ ℱ. The intersection of ℱ is the set of objects belonging to every element of ℱ, and we write it ⋂ ℱ. So the inclusion criterion for ⋃ ℱ is that x ∈ X for at least one X ∈ ℱ, and the inclusion criterion for ⋂ ℱ is that x ∈ X for every X ∈ ℱ:

Definition 15 (Union of a collection). ⋃ ℱ = {x : x ∈ X for at least one X ∈ ℱ}.

Figure 22. The union of a collection ℱ = {X₁, X₂, X₃} is the whole shaded region.

Definition 16 (Intersection of a collection). ⋂ ℱ = {x : x ∈ X for every X ∈ ℱ}.

Figure 23. The intersection of a collection ℱ = {X₁, X₂, X₃, X₄} is the shaded region.

For example, ⋃ {{7, 8}, {8, 9}, {9, 10}} = {7, 8, 9, 10}. On the other hand, ⋂ {{7, 8}, {8, 9}, {9, 10}} = ∅, since no number is in all three. And ⋂ {{7, 8, 9}, {8, 9}, {9, 10}} = {9}.

Taking ℱ to be {X, Y} gives us back the operations we already have: ⋃ {X, Y} = X ∪ Y and ⋂ {X, Y} = X ∩ Y. We will always assume that ℱ is non-empty when taking an intersection.

Proposition 45 (each set is a subset of the union of its collection). For any collection ℱ and any X ∈ ℱ, X ⊂ ⋃ ℱ.

Suppose that x ∈ X. By the definition of union, for any Y ∈ ℱ, if y ∈ Y then y ∈ ⋃ ℱ. In particular, since X ∈ ℱ, we have x ∈ ⋃ ℱ.

Proposition 46 (the intersection of a collection is a subset of each). For any collection ℱ and any X ∈ ℱ, ⋂ ℱ ⊂ X.

Suppose that x ∈ ⋂ ℱ. Then, by the definition of intersection, for each Y ∈ ℱ, we have x ∈ Y. In particular, since X ∈ ℱ, we have x ∈ X.

These are the general versions of Propositions 15 (each set is a subset of the union) and 8 (an intersection is a subset of each set).

In practice you will often see these written with an index. If we have a set X_i for each i in some set I, people write ⋃_{i ∈ I} X_i for ⋃ {X_i : i ∈ I}, and ⋂_{i ∈ I} X_i for ⋂ {X_i : i ∈ I}. This is just a way of labelling the elements of the collection, and nothing new is going on.