9

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? $$ \mathbb{R}_{+} = \{(x, y)\in \mathbb{R}^2: x, y \geq 0 \} $$

MMM
  • 751
  • Note that the linked question does cover both positive and non-negative reals. –  May 11 '18 at 08:59

4 Answers4

9

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.

Daniel Robert-Nicoud
  • 29,795
  • 2
    This notation has the virtue of being self-explanatory. Also, the subscript doesn't interfere with any dimension indicator that would normally attach to the $\Bbb R$ as a supersccript. – John Bentin Jul 08 '18 at 21:09
3

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.

YukiJ
  • 2,529
  • Thank you! What about two dimensional case? – MMM May 11 '18 at 08:54
  • @JonasAl-Hadad Just added that, sorry! – YukiJ May 11 '18 at 08:55
  • Ok, thanks! I just wonder about cases: ${(x,y)\in\mathbb{R}^2: x, y\geq 0}$ and ${(x,y)\in\mathbb{R}^2: x, y>0}$. I think that in both cases symbol $\mathbb{R}_{+}^2$ can be used. But I would like to have two different symbols ;) – MMM May 11 '18 at 09:01
  • @JonasAl-Hadad Why not use $\mathbb{R}^2_{>0}$ for the latter case to avoid misinterpretation? – YukiJ May 11 '18 at 09:05
  • Ok, so I think the best way is to use: $\mathbb{R}{0}^2$ and $\mathbb{R}{+}^2$ – MMM May 11 '18 at 09:08
1

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}

Bernard
  • 175,478
0

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 $\times$.

Allawonder
  • 13,327