Khanh at 18:40, 4 February 2022

2022-02-04T18:40:11Z

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A set of sets is usually referred to as a ''family'' or ''collection'' of sets. Often, families of sets are written with either a script or [http://en.wikipedia.org/wiki/Fraktur_(script) Fraktur] font to easily distinguish them from other sets. For a family of sets <math>\mathfrak{A}</math> , define the ''union'' and ''intersection'' of the family by,

:{{math|size=1.4em|<math>\begin{align}\bigcup\mathfrak{A}&=\bigcup_{A\in\mathfrak A}A=\{x:x\text{ is in some }A\in\mathfrak A\}\\\bigcap\mathfrak{A}&=\bigcap_{A\in\mathfrak A}A=\{x:x\text{ is in all }A\in\mathfrak A\}\end{align}</math>}}

For a family of sets, we say that it is ''pairwise disjoint'' if any two distinct sets we choose from the family are disjoint.

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 109-115

← Older revision		Revision as of 16:54, 1 October 2021
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	==Resources==		==Resources==
	* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 109-115		* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 109-115
		+	* [https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/5%3A_Set_Theory/5.5%3A_Indexed_Families_of_Sets Indexed Families of Sets], Mathematics LibreTexts

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-A set of sets is usually referred to as a ''family'' or ''collection'' of sets. Often, families of sets are written with either a script or [http://en.wikipedia.org/wiki/Fraktur_(script) Fraktur] font to easily distinguish them from other sets. For a family of sets <math>\mathfrak{A}</math> , define the ''union'' and ''intersection'' of the family by,
+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''' or a '''set-family''' or a '''set-system'''.
-:{{math|size=1.4em|<math>\begin{align}\bigcup\mathfrak{A}&=\bigcup_{A\in\mathfrak A}A=\{x:x\text{ is in some }A\in\mathfrak A\}\\\bigcap\mathfrak{A}&=\bigcap_{A\in\mathfrak A}A=\{x:x\text{ is in all }A\in\mathfrak A\}\end{align}</math>}}
+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.
-For a family of sets, we say that it is ''pairwise disjoint'' if any two distinct sets we choose from the family are disjoint.
+A finite family of subsets of a finite set S is also called a '''hypergraph'''.
 ==Resources==
 * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 109-115
 * [https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/5%3A_Set_Theory/5.5%3A_Indexed_Families_of_Sets Indexed Families of Sets], Mathematics LibreTexts

Sets:Families - Revision history

Khanh at 18:40, 4 February 2022

Lila: /* Resources */

Lila: Created page with "A set of sets is usually referred to as a ''family'' or ''collection'' of sets. Often, families of sets are written with either a script or [http://en.wikipedia.org/wiki/Frakt..."