Talk:Floor function
From Maths
Phrasing the characteristic property
I've proved on paper that given [ilmath]\text{Floor}:\mathbb{R}_{\ge 0}\rightarrow\mathbb{N}_{\ge 0} [/ilmath] and property 3 on the previous page (the epsilon one) that properties 1 and 2 follow as well as a corollary - getting to the corollary is why I'm bothering with this.
Do I keep the characteristic property as is? Or do I adjust it to include the function statement? Both or do I combine them like below:
- Characteristic property:[ilmath]\newcommand{\Floor}[1]{ {\text{Floor}{\left({#1}\right)} } } [/ilmath]
- [math]\Big(\forall x\in\mathbb{R}_{\ge 0}\big[\Floor{x}\in\mathbb{N}_{\ge 0}\subseteq\mathbb{R}_{\ge 0}\big]\Big)\wedge\Big(\forall x\in\mathbb{R}_{\ge 0}\exists\epsilon\in[0,1)\subseteq\mathbb{R}_{\ge 0}\big[\Floor{x}+\epsilon\eq x\big]\Big)[/math]
- So
- [math]\forall x\in\mathbb{R}_{\ge 0}\Big[\big(\Floor{x}\in\mathbb{N}_{\ge 0}\subseteq\mathbb{R}_{\ge 0}\big)\wedge\big(\exists\epsilon\in[0,1)\subseteq\mathbb{R}_{\ge 0}[\Floor{x}+\epsilon\eq x]\big)\Big][/math]
