> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/400-abstract-algebra/terms/field-of-sets.md).

# Field of sets

<https://en.wikipedia.org/wiki/Field_of_sets>

A **field of sets** is a mathematical structure consisting of a pair ⟨X, 𝓕⟩ where X is a set and 𝓕 is a family of subsets of X called an algebra over X that contains *the empty set* as an element (identity), and is closed under the operations of taking *complements* in X, finite *unions*, and finite *intersections*.

Equivalently, an algebra over X is a subset 𝓕 of the power set of X such that

* ∅ ∈ 𝓕              (𝓕 has the identity element)
* ∀A ∈ 𝓕. A' ∈ 𝓕     (𝓕 is closed under compliment)
* ∀AB ∈ 𝓕. A  B ∈ 𝓕 (𝓕 is closed under intersection)
* ∀AB ∈ 𝓕. A ∪ B ∈ 𝓕 (𝓕 is closed under union)