[ilmath]\mathbb{R}_+[/ilmath] (notation)
Notation
- Do not use this notation
The notation [ilmath]\mathbb{R}_+[/ilmath] is horribly ambiguous, it could easily mean either of the following:
- [ilmath]\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}[/ilmath]
- Which is used by^{[1]}
- [ilmath]\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x > 0\}[/ilmath]
TODO: Find reference for this
Solution
Use [ilmath]\mathbb{R}_{\ge 0} [/ilmath], this requires one more letter and there is literally no way to interpret this wrongly. Both [ilmath]\mathbb{R}_{\ge 0} [/ilmath] and [ilmath]\mathbb{R}_{> 0} [/ilmath] are extremely clear. Furthermore this notation extends naturally to things like [ilmath]\mathbb{R}_{\ge 5} [/ilmath] or [ilmath]\mathbb{R}_{\le 0} [/ilmath], it is obvious what these mean. See Denoting commonly used subsets of R
Ambiguities
- positive ([ilmath]+[/ilmath]) means [ilmath]>0[/ilmath] and we use non-negative to mean [ilmath]\ge 0[/ilmath] (as this is literally not negative)
Problems
- If we use [ilmath]\mathbb{R}_+[/ilmath] to denote [ilmath]\ge 0[/ilmath] how would we denote [ilmath]>0[/ilmath]?
Conflicts
When we use [ilmath]\mathbb{N}_+[/ilmath] it is pretty clear that this means the set [ilmath]\{1,2,\ldots,n,\ldots\} [/ilmath], that is - this conflicts with definition 1) above and thus using [ilmath]\mathbb{R}_+[/ilmath] as [ilmath]\mathbb{R}_{\ge 0} [/ilmath] violates the Doctrine of least surprise