Difference between revisions of "Family of Sets"

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A [[Family of Sets]] <math>F</math> is a [[set]] composed of [[subset]]s from [[set]] <math>S</math>.
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A [[Family of Sets]] is a [[set]] composed of [[subset]]s from a [[set]].
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* <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.  
(* Let ''S'' = {a,b,c,1,2}, an example of a family of sets over ''S'' (in the multiset sense) is given by ''F'' = {A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub>, A<sub>4</sub>} where A<sub>1</sub> = {a,b,c}, A<sub>2</sub> = {1,2}, A<sub>3</sub> = {1,2} and A<sub>4</sub> = {a,b,1}.
 
 
** 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>See:</B> [[Set Theory]], [[Subset]], [[Class (Set Theory)]].
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** a [[Sperner Set Family]],
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** a [[Helly Set Family]].
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* <B>Counter-Example(s):</B>
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** a [[Subset]].
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* <B>See:</B> [[Set Theory]], [[Class (Set Theory)]].
 
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== References ==
 
== References ==
  
=== 2014 ===
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=== 2019 ===
* (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Family_of_sets Retrieved:2014-4-21.
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* (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</B> of ''S'', or a '''family of sets</B> over ''S''. More generally, a collection of any sets whatsoever is called a '''family of sets</B>. <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.
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** 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__
 
=== 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 [[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.
 

Revision as of 23:19, 10 November 2019

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



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.