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...") |
m |
||
| Line 16: | Line 16: | ||
| 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: [ilmath]\sigma[/ilmath]-finite is under "finite".
- The specific case contains extra information, so [ilmath]\sigma[/ilmath]-finite is under finite, but specifically [ilmath]\sigma[/ilmath]-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 |
| [ilmath]\backslash[/ilmath]-closed[1] | CLOSED_backslash | To say [ilmath]\mathcal{A} [/ilmath] is [ilmath]\backslash[/ilmath]-closed uses [ilmath]\backslash[/ilmath] to denote set subtraction[Note 1], this means [math]\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}][/math] | |
| [ilmath]\cap[/ilmath]-closed[1] | CLOSED_cap | If [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed then [math]\forall A,B\in\mathcal{A}[A\cap B\in\mathcal{A}][/math] - [ilmath]\mathcal{A} [/ilmath] is closed under finite intersection | |
| [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed[1] | CLOSED_cap_sigma | closed under 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[1] | CLOSED_complement | If [ilmath]\mathcal{A} [/ilmath] is closed under complement then [math]\forall A\in\mathcal{A}[A^c\in\mathcal{A}][/math] | |
| [ilmath]\cup[/ilmath]-closed[1] | CLOSED_cup | If [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed then [math]\forall A,B\in\mathcal{A}[A\cup B\in\mathcal{A}][/math] - [ilmath]\mathcal{A} [/ilmath] is closed under finite union | |
| [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[1] | CLOSED_cup_sigma | closed under countably infinite union. [math]\forall (A_n)_{n=1}^\infty\subseteq\mathcal{A}[\cup_{n=1}^\infty A_n\in\mathcal{A}][/math] | |
| [ilmath]\backslash[/ilmath]-closed | CLOSED_division | See CLOSED_backslash
|
Notes
- ↑ This is because [ilmath]-[/ilmath]-closed is not a good way to write this
