What symbol should I use for nonnegative real numbers

Question

I know that positive real numbers can be denoted in the following way: $$ \mathbb{R}^{+} = \{x \in \mathbb{R}: x>0\} $$

I also encounter this symbol: $$ \mathbb{R}^{>0} = \{x \in \mathbb{R}: x>0\} $$

My question is:

1. How can I denote nonnegative real numbers, theoretically it should be $\mathbb{R}^{\geq 0}$, but I think it is not elegant way.

2. I have the same question about two dimensional case. I met the following notation. Is it correct? $$ \mathbb{R}_{+} = \{(x, y)\in \mathbb{R}^2: x, y \geq 0 \} $$

Answer 1

I often see people use $\mathbb{R}_0^+$.

Concerning your second question: it would be better to add the superscript 2: $\mathbb{R}^2_+$. Otherwise it is not necessarily clear that the $\mathbb{R}^2$ is referred to.

Answer 2

I've usually seen (and use): \begin{align} \mathbf R_+^{\phantom{*}}=\{\,x\in\mathbf R\mid x\ge 0\,\}\\[1ex] \mathbf R_+^*=\{\,x\in\mathbf R\mid x > 0\,\} \end{align}

Answer 3

You can use any compact notation of your choice as long as you define it well. Suppose, for example, that I wish to use $\mathcal R$ to denote the nonnegative reals, then since $\mathbb R^+$ is a fairly well-known notation for the positive reals, I can just say, Let $$\mathcal R = \mathbb R^+ \cup \{0\}.$$

Something similar can be done for any $n$-dimensional euclidean space, where you wish to deal with the members in the first $2^n$-ant of the space and the origin $(0,0,0,0...)$, where there are $n$ zeros. Just pick a suitable compact notation and define it beforehand by using the set operation $\cup$, or more simply just use $\mathcal R$ as defined above and the cartesian product $\times$.

Answer 4

I have often seen $\mathbb{R}_{\ge0}$.

In the end, it doesn't really matter as long as the notation you use is not too heavy, and you define it well.