Difference between revisions of "Algebra of sets"

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{{:Algebra of sets/Infobox}}
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: '''Note: ''' Every ''algebra of sets'' is a ''[[ring of sets]]'' (see below)
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__TOC__
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==Definition==
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An ''algebra of sets'' is a collection of sets, {{M|\mathcal{A} }} such that{{rMTH}}:
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* {{M|1=\forall A\in\mathcal{A}[A^C\in\mathcal{A}]}}<ref group="Note">Recall {{M|1=A^C:=X-A}} - the [[complement]] of {{M|A}} in {{M|X}}</ref>
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** In words: For all {{M|A}} in {{M|\mathcal{A} }} the [[complement]] of {{M|A}} (with respect to {{M|X}}) is also in {{M|\mathcal{A} }}
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* {{M|1=\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]}}
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** In words: For all {{M|A}} and {{M|B}} in {{M|\mathcal{A} }} their [[union]] is also in {{M|\mathcal{A} }}
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'''[[Algebra of sets#Proof of claims|Claim 1]]: ''' Every ''algebra of sets'' is also a ''[[ring of sets]]''
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==Immediate properties==
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{{Todo|Do this as a list of inline theorem boxes}}
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* {{M|\mathcal{A} }} is {{M|\setminus}}-closed
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* {{M|\emptyset\in\mathcal{A} }}
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* {{M|X\in\mathcal{A} }}
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* {{M|\mathcal{A} }} is {{M|\cap}}-closed
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==Proof of claims==
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{{Begin Inline Theorem}}
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'''Claim 1: ''' Every ''algebra of sets'' is also a ''[[ring of sets]]''
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{{Begin Inline Proof}}
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This is trivial. In order to show {{M|\mathcal{A} }} is a [[ring of sets]] we require two properties:
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# {{M|\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]}} - this is clearly satisfied by definition of an algebra of sets
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# {{M|\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}} - that is {{M|\mathcal{A} }} must be {{M|\setminus}}-closed
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#* But we've already shown this in the [[Algebra of sets#Immediate properties|''immediate properties'']] section above!
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This completes the proof
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{{End Proof}}{{End Theorem}}
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==See also==
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* [[Types of set algebras]]
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* [[ring of sets]]
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** [[sigma-ring|{{sigma|ring}}]]
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* [[sigma-algebra|{{sigma|algebra}}]]
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
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{{Measure theory navbox|plain}}
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{{Definition|Measure Theory}}
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=OLD PAGE=
 
An Algebra of sets is sometimes called a '''Boolean algebra'''
 
An Algebra of sets is sometimes called a '''Boolean algebra'''
  
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We will show later that every Algebra of sets is an [[Algebra of sets]]
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We will show later that every Algebra of sets is an [[Algebra of sets]] {{Todo|what could this mean?}}
  
 
==Definition==
 
==Definition==

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Algebra of sets
[ilmath]\mathcal{A}\subseteq\mathcal{P}(X)[/ilmath]
For an algebra of sets, [ilmath]\mathcal{A} [/ilmath] on [ilmath]X[/ilmath]
Defining properties:
1) [ilmath]\forall A\in\mathcal{A}[A^C\in\mathcal{A}][/ilmath]
2) [ilmath]\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A} ][/ilmath]
Note: Every algebra of sets is a ring of sets (see below)

Definition

An algebra of sets is a collection of sets, [ilmath]\mathcal{A} [/ilmath] such that[1]:

  • [ilmath]\forall A\in\mathcal{A}[A^C\in\mathcal{A}][/ilmath][Note 1]
    • In words: For all [ilmath]A[/ilmath] in [ilmath]\mathcal{A} [/ilmath] the complement of [ilmath]A[/ilmath] (with respect to [ilmath]X[/ilmath]) is also in [ilmath]\mathcal{A} [/ilmath]
  • [ilmath]\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}][/ilmath]
    • In words: For all [ilmath]A[/ilmath] and [ilmath]B[/ilmath] in [ilmath]\mathcal{A} [/ilmath] their union is also in [ilmath]\mathcal{A} [/ilmath]

Claim 1: Every algebra of sets is also a ring of sets

Immediate properties

TODO: Do this as a list of inline theorem boxes

  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\setminus[/ilmath]-closed
  • [ilmath]\emptyset\in\mathcal{A} [/ilmath]
  • [ilmath]X\in\mathcal{A} [/ilmath]
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed

Proof of claims

Claim 1: Every algebra of sets is also a ring of sets

This is trivial. In order to show [ilmath]\mathcal{A} [/ilmath] is a ring of sets we require two properties:

  1. [ilmath]\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}][/ilmath] - this is clearly satisfied by definition of an algebra of sets
  2. [ilmath]\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}][/ilmath] - that is [ilmath]\mathcal{A} [/ilmath] must be [ilmath]\setminus[/ilmath]-closed

This completes the proof

See also

Notes

  1. ↑ Recall [ilmath]A^C:=X-A[/ilmath] - the complement of [ilmath]A[/ilmath] in [ilmath]X[/ilmath]

References

  1. ↑ Measure Theory - Paul R. Halmos

v • d • e
 
Measure Theory
Overview of the basic and important objects studied in measure theory
Set-structures
Measures
Pre-measure [ilmath]\leadsto[/ilmath] outer-measure [ilmath]\leadsto[/ilmath] measure (see Extending pre-measures to measures (via: Extending pre-measures to outer-measures [ilmath]\rightarrow[/ilmath] "Restricting outer-measures to measures" maybe?). Alternatively: inner-measure can be used.
measure space
Functions
Spaces
Integrals

OLD PAGE

An Algebra of sets is sometimes called a Boolean algebra

We will show later that every Algebra of sets is an Algebra of sets

TODO: what could this mean?


Definition

An class [ilmath]R[/ilmath] of sets is an Algebra of sets if[1]:

  • [math][A\in R\wedge B\in R]\implies A\cup B\in R[/math]
  • [math]A\in R\implies A^c\in R[/math]

So an Algebra of sets is just a Ring of sets containing the entire set it is a set of subsets of!

Every Algebra is also a Ring

Since for [math]A\in R[/math] and [math]B\in R[/math] we have:

[math]A-B=A\cap B' = (A'\cup B)'[/math] we see that being closed under Complement and Union means it is closed under Set subtraction

Thus it is a Ring of sets

See also

References

  1. ↑ p21 - Halmos - Measure Theory - Graduate Texts In Mathematics - Springer - #18
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