# Difference between revisions of "Family of Sets"

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− | A [[Family of Sets]] | + | A [[Family of Sets]] is a [[set]] composed of [[subset]]s from a [[set]]. |

+ | * <B>AKA:</B> [[Set of Sets]], [[Set Set]]. | ||

* <B>Context:</B> | * <B>Context:</B> | ||

** It can be [[represented by]] a [[Family of Sets Data Structure]]. | ** It can be [[represented by]] a [[Family of Sets Data Structure]]. | ||

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** a [[Power Set]] '''P'''(''S'') is a family of sets over ''S''. | ** a [[Power Set]] '''P'''(''S'') is a family of sets over ''S''. | ||

** a [[finite set|''k</i>-subsets]] ''S''<sup>(''k'')</sup> of a set ''S'' form a family of sets. | ** a [[finite set|''k</i>-subsets]] ''S''<sup>(''k'')</sup> of a set ''S'' form a family of sets. | ||

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** The class Ord of all [[ordinal number]]s is a ''large'' family of sets; that is, it is not itself a set but instead a [[Class (set theory)|proper class]]. | ** The class Ord of all [[ordinal number]]s is a ''large'' family of sets; that is, it is not itself a set but instead a [[Class (set theory)|proper class]]. | ||

− | * <B> | + | ** a [[Sperner Set Family]], |

+ | ** a [[Helly Set Family]]. | ||

+ | * <B>Counter-Example(s):</B> | ||

+ | ** a [[Subset]]. | ||

+ | * <B>See:</B> [[Set Theory]], [[Class (Set Theory)]]. | ||

---- | ---- | ||

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== References == | == References == | ||

− | === | + | === 2019 === |

− | * (Wikipedia, | + | * (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Family_of_sets Retrieved:2019-11-10. |

− | ** In [[set theory]] and related branches of [[mathematics]], a collection ''F'' of [[subset]]s of a given [[Set (mathematics)|set]] ''S'' is called a '''family of subsets | + | ** In [[set theory]] and related branches of [[mathematics]], a collection ''F'' of [[subset]]s of a given [[Set (mathematics)|set]] ''S'' is called a '''family of subsets''' of ''S'', or a '''family of sets''' over ''S''. More generally, a collection of any sets whatsoever is called a '''family of sets'''. <P> 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 [[Class (set theory)|proper class]] rather than a set. |

<references/> | <references/> | ||

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[[Category:Concept]] | [[Category:Concept]] | ||

__NOTOC__ | __NOTOC__ | ||

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## Revision as of 23:19, 10 November 2019

A Family of Sets is a set composed of subsets from a set.

**AKA:**Set of Sets, Set Set.**Context:**- It can be represented by a Family of Sets Data Structure.

**Example(s):**- a Power Set
**P**(*S*) is a family of sets over*S*. - a
*k*-subsets*S*^{(k)}of a set*S*form a family of sets. - 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. - a Sperner Set Family,
- a Helly Set Family.

- a Power Set
**Counter-Example(s):**- a Subset.

**See:**Set Theory, Class (Set Theory).

## References

### 2019

- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Family_of_sets Retrieved:2019-11-10.
- In set theory and related branches of mathematics, 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 any sets whatsoever is called a**family of sets**.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.

- In set theory and related branches of mathematics, a collection