Difference between revisions of "Floor function"
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This is a corollary to property 3 coupled with the definition (domain and co domain) of the floor [[function]]: | This is a corollary to property 3 coupled with the definition (domain and co domain) of the floor [[function]]: | ||
* {{M|\forall x\in\mathbb{R}_{\ge 0}\big[\Floor{x}\le x<\Floor{x}+1\big]}} | * {{M|\forall x\in\mathbb{R}_{\ge 0}\big[\Floor{x}\le x<\Floor{x}+1\big]}} | ||
| + | This statement is a critical part of finding [[Mdm]]s and was used in: | ||
| + | * [[Mdm of a discrete distribution lemma]] | ||
| + | ** Which at the time of writing ([[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:20, 21 January 2018 (UTC)) only exists as notes: [[Notes:Mdm of a discrete distribution lemma]] | ||
{{Definition|Analysis|Real Analysis}} | {{Definition|Analysis|Real Analysis}} | ||
Latest revision as of 21:20, 21 January 2018
[ilmath]\newcommand{\Floor}[1]{ {\text{Floor}{\left({#1}\right)} } } [/ilmath]
Contents
Definition
For [ilmath]x\in\mathbb{R}_{\ge 0} [/ilmath] there is no variation on the meaning of the floor function, however for negative numbers there are varying conventions.
Non-negative
Defined as follows:
- [ilmath]\text{Floor}:\mathbb{R}_{\ge 0}\rightarrow\mathbb{N}_0[/ilmath] by [ilmath]\text{Floor}:x\mapsto[/ilmath][ilmath]\text{Max} [/ilmath][ilmath](T_x)[/ilmath] where [ilmath]T_x:\eq\big\{n\in\mathbb{N}_0\ \big\vert\ n\le x\big\}\subseteq\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0} [/ilmath] - note that the maximum element is defined as [ilmath]T_x[/ilmath] is always finite.
- This has the property that [ilmath]x\le\Floor{x} [/ilmath].
Negative numbers
Researching this opened my eyes to a massive dispute.... consensus seems to be that [ilmath]x\le \Floor{x} [/ilmath] is maintained, rounding is a separate and massive issue!
References
Future work
Properties
- [ilmath]\forall n\in\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0}\big[\Floor{n}\eq n\big][/ilmath], or [ilmath]\text{Floor}\vert_{\mathbb{N}_0}\eq\text{Id}_{\mathbb{N}_0} [/ilmath] - its restriction to [ilmath]\mathbb{N}_0[/ilmath] is the identity map on [ilmath]\mathbb{N}_0[/ilmath]
- [ilmath]\forall x,y\in\mathbb{R}_{\ge 0}\big[(x\le y)\implies\big(\Floor{x}\le\Floor{y}\big)\big][/ilmath] - monotonicity
- [ilmath]\forall x\in\mathbb{R}_{\ge 0}\exists\epsilon\in[0,1)\subseteq\mathbb{R}\big[x\eq\Floor{x}+\epsilon\big][/ilmath] - the characteristic property of the floor function
I believe that [ilmath]3\implies 1[/ilmath] and [ilmath]3\implies 2[/ilmath] might be possible, so these are perhaps in the wrong order. I just wanted to write down some notes before they get put into the massive stack of unfiled paper
This is a corollary to property 3 coupled with the definition (domain and co domain) of the floor function:
- [ilmath]\forall x\in\mathbb{R}_{\ge 0}\big[\Floor{x}\le x<\Floor{x}+1\big][/ilmath]
This statement is a critical part of finding Mdms and was used in:
- Mdm of a discrete distribution lemma
- Which at the time of writing (Alec (talk) 21:20, 21 January 2018 (UTC)) only exists as notes: Notes:Mdm of a discrete distribution lemma
