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

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

  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.