Difference between revisions of "Index of properties"
From Maths
(Created page with "==Index== Note: * Things are indexed by the ''adjective in the property'', for example: {{Sigma|finite}} is under "finite". * The specific case contains extra information, so...") |
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| To say something is ''closed under'' means one cannot leave it through the stated property, eg "the integers are ''closed under'' addition | | To say something is ''closed under'' means one cannot leave it through the stated property, eg "the integers are ''closed under'' addition | ||
|- | |- | ||
| − | | {{M|\backslash}}-closed | + | | {{M|\backslash}}-closed<ref name="PTACC">Probability Theory - A comprehensive course - Second Edition - Achim Klenke</ref> |
| CLOSED_backslash | | CLOSED_backslash | ||
| To say {{M|\mathcal{A} }} is {{M|\backslash}}-closed uses {{M|\backslash}} to denote [[Set subtraction|set subtraction]]<ref group="Note">This is because {{M|-}}-closed is not a good way to write this</ref>, this means <math>\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]</math> | | To say {{M|\mathcal{A} }} is {{M|\backslash}}-closed uses {{M|\backslash}} to denote [[Set subtraction|set subtraction]]<ref group="Note">This is because {{M|-}}-closed is not a good way to write this</ref>, this means <math>\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]</math> | ||
|- | |- | ||
| − | | {{M|\cap}}-closed | + | | {{M|\cap}}-closed<ref name="PTACC"/> |
| CLOSED_cap | | CLOSED_cap | ||
| If {{M|\mathcal{A} }} is ''{{M|\cap}}-closed'' then <math>\forall A,B\in\mathcal{A}[A\cap B\in\mathcal{A}]</math> - {{M|\mathcal{A} }} is closed under finite [[Intersection|intersection]] | | If {{M|\mathcal{A} }} is ''{{M|\cap}}-closed'' then <math>\forall A,B\in\mathcal{A}[A\cap B\in\mathcal{A}]</math> - {{M|\mathcal{A} }} is closed under finite [[Intersection|intersection]] | ||
|- | |- | ||
| − | | {{Sigma|{{M|\cap}}-closed}} | + | | {{Sigma|{{M|\cap}}-closed}}<ref name="PTACC"/> |
| CLOSED_cap_sigma | | CLOSED_cap_sigma | ||
| closed under [[Countably infinite|countably infinite]] intersection. <math>\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cap_{n=1}^\infty A_n\in\mathcal{A}]</math> | | closed under [[Countably infinite|countably infinite]] intersection. <math>\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cap_{n=1}^\infty A_n\in\mathcal{A}]</math> | ||
|- | |- | ||
| − | | closed under complement | + | | closed under complement<ref name="PTACC"/> |
| CLOSED_complement | | CLOSED_complement | ||
| If {{M|\mathcal{A} }} is closed under [[Complement|complement]] then <math>\forall A\in\mathcal{A}[A^c\in\mathcal{A}]</math> | | If {{M|\mathcal{A} }} is closed under [[Complement|complement]] then <math>\forall A\in\mathcal{A}[A^c\in\mathcal{A}]</math> | ||
|- | |- | ||
| − | | {{M|\cup}}-closed | + | | {{M|\cup}}-closed<ref name="PTACC"/> |
| CLOSED_cup | | CLOSED_cup | ||
| If {{M|\mathcal{A} }} is ''{{M|\cup}}-closed'' then <math>\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]</math> - {{M|\mathcal{A} }} is closed under finite [[Union|union]] | | If {{M|\mathcal{A} }} is ''{{M|\cup}}-closed'' then <math>\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}]</math> - {{M|\mathcal{A} }} is closed under finite [[Union|union]] | ||
|- | |- | ||
| − | | {{Sigma|{{M|\cup}}-closed}} | + | | {{Sigma|{{M|\cup}}-closed}}<ref name="PTACC"/> |
| CLOSED_cup_sigma | | CLOSED_cup_sigma | ||
| closed under [[Countably infinite|countably infinite]] union. <math>\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cup_{n=1}^\infty A_n\in\mathcal{A}]</math> | | closed under [[Countably infinite|countably infinite]] union. <math>\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cup_{n=1}^\infty A_n\in\mathcal{A}]</math> | ||
Latest revision as of 20:43, 15 June 2015
Index
Note:
- Things are indexed by the adjective in the property, for example: σ-finite is under "finite".
- The specific case contains extra information, so σ-finite is under finite, but specifically σ-finite
- The word "under" is ignored in the index
| Adjective | Specific case | Index | Description |
|---|---|---|---|
| Closed | (general) | CLOSED | To say something is closed under means one cannot leave it through the stated property, eg "the integers are closed under addition |
| ∖-closed[1] | CLOSED_backslash | To say A is ∖-closed uses ∖ to denote set subtraction[Note 1], this means ∀A,B∈A[A−B∈A] | |
| ∩-closed[1] | CLOSED_cap | If A is ∩-closed then ∀A,B∈A[A∩B∈A] - A is closed under finite intersection
| |
| σ-∩-closed[1] | CLOSED_cap_sigma | closed under countably infinite intersection. ∀(An)∞n=1⊆A[∩∞n=1An∈A] | |
| closed under complement[1] | CLOSED_complement | If A is closed under complement then ∀A∈A[Ac∈A] | |
| ∪-closed[1] | CLOSED_cup | If A is ∪-closed then ∀A,B∈A[A∪B∈A] - A is closed under finite union
| |
| σ-∪-closed[1] | CLOSED_cup_sigma | closed under countably infinite union. ∀(An)∞n=1⊆A[∪∞n=1An∈A] | |
| ∖-closed | CLOSED_division | See CLOSED_backslash
|
Notes
- Jump up ↑ This is because −-closed is not a good way to write this
