Types of set algebras/Type table
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Contents
Type table
System Type | Definition | Deductions |
---|---|---|
Ring[1][2] |
| |
[ilmath]\sigma[/ilmath]-ring[1][2] |
|
|
Algebra[1][2] |
|
|
[ilmath]\sigma[/ilmath]-algebra[1][2] |
|
|
Semiring[1] |
TODO: Page 3 in[1] | |
Dynkin system[1][3] |
|
|
Theorems
- ↑ 1.0 1.1 Using Class of sets closed under set-subtraction properties we know that if [ilmath]\mathcal{A} [/ilmath] is closed under Set subtraction then:
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed
- [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[ilmath]\implies[/ilmath][ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed
- ↑ Using Class of sets closed under complements properties we see that if [ilmath]\mathcal{A} [/ilmath] is closed under complements then:
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed
Notes
- ↑ Closed under finite Set subtraction
- ↑ Closed under finite Union
- ↑ As given [ilmath]A\in\mathcal{A} [/ilmath] we must have [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath]
- ↑ closed under finite or countably infinite union
- ↑ Note that [ilmath]A-B=A\cap B^c=(A^c\cup B)^c[/ilmath] - or that [ilmath]A-B=(A^c\cup B)^c[/ilmath] - so we see that being closed under union and complement means we have closure under set subtraction.
- ↑ As we are closed under set subtraction we see [ilmath]A-A=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{A} [/ilmath]
- ↑ As we are closed under set subtraction we see that [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath], so [ilmath]\emptyset\in\mathcal{A} [/ilmath] - but we are also closed under complements, so [ilmath]\emptyset^c\in\mathcal{A} [/ilmath] and [ilmath]\emptyset^c=\Omega\in\mathcal{A}[/ilmath]
- ↑ Trivial - satisfies the definitions
- ↑ As [ilmath]\Omega^c=\emptyset[/ilmath] by being closed of complements, [ilmath]\emptyset\in\mathcal{A} [/ilmath]