> 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/200-set-theory/topics/family-of-sets.md).

# Family of sets

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

A collection `F` of subsets of a given set `S` is called a *family of subsets* of `S`, or a *family of sets over `S`*.

More generally, a collection of sets is called a *family of sets* or a *set-family* or a *set-system* (as long as that collection contains only sets as members).

The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a *proper class* rather than a set.

A finite family of subsets of a finite set `S` is also called a *hypergraph*.

Examples of set families:

* The powerset 𝓟(S) is a family of sets over S
* The k-subsets S⁽ᵏ⁾ of a set S form a family of sets.
* Let S = {a,b,c,1,2}, an example of a family of sets over S&#x20;

  (in the multiset sense) is given by F = {A1, A2, A3, A4} where&#x20;

  A1 = {a,b,c}, A2 = {1,2}, A3 = {1,2} and A4 = {a,b,1}
* The class `Ord` of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a proper class.

Properties:

* Any family of subsets of S is itself a subset of the power set 𝓟(S) if it has no repeated members.
* Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe).
* *Hall's marriage theorem*, due to Philip Hall, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a system of distinct representatives.

## Families

(from "Naive Set Theory" by Halamos 1960)

Suppose a function `f : I → X`; then\
an element of the domain `I` is called an *index*,\
and the set `I` is called the *index set*,\
the range of the function is called an *indexed set*,\
the function itself is called a *family*,\
and the value of the function `f` at an index `i`\
is called a *term of the family* and denoted by `fᵢ`.