[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:

1. [ilmath]\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}[/ilmath]
• Which is used by[1]
2. [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