$\mathbb{R}_+$ (notation)

Notation

Do not use this notation

The notation $\mathbb{R}_+$ is horribly ambiguous, it could easily mean either of the following:

1. $\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}$
   • Which is used by
2. $\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x > 0\}$

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TODO: Find reference for this

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Solution

Use $\mathbb{R}_{\ge 0}$, this requires one more letter and there is literally no way to interpret this wrongly. Both $\mathbb{R}_{\ge 0}$ and $\mathbb{R}_{> 0}$ are extremely clear. Furthermore this notation extends naturally to things like $\mathbb{R}_{\ge 5}$ or $\mathbb{R}_{\le 0}$, it is obvious what these mean. See Denoting commonly used subsets of R

Ambiguities

• positive ($+$) means $>0$ and we use non-negative to mean $\ge 0$ (as this is literally not negative)

Problems

• If we use $\mathbb{R}_+$ to denote $\ge 0$ how would we denote $>0$?

Conflicts

When we use $\mathbb{N}_+$ it is pretty clear that this means the set $\{1,2,\ldots,n,\ldots\}$, that is - this conflicts with definition 1) above and thus using $\mathbb{R}_+$ as $\mathbb{R}_{\ge 0}$ violates the Doctrine of least surprise