What does $\mathbb{R}_0^+$ mean

discrete mathematicsnotation

What does $\mathbb{R}_0^+$ mean?

I know that the superscript indicates all real numbers $\gt0$ but what does the subscript indicate in set notation?

Thanks!

EDIT: The context is $f:\mathbb R\to\mathbb R_0^+,f(x)=x^2$ so I'm assuming it means including $0$? Does this apply to general notation as well?

Best Answer

For you background, there is a discrepancy between notations in the U.S and some countries in Europe (mainly France and Italy, but not only). This is of course very general, there may be exceptions, but it explains why at some point the need for an unambiguous notation appeared.

Rem: I use U.S and E.U below, but again it may vary from people to people, don't be offended if you do differently...

  • $\mathbb N$ in U.S does not contains $0$ while contains $0$ in E.U

Thus the unambiguous notation is $\mathbb N_0$ for $\{0,1,2,\cdots\}$.

In E.U we use $\mathbb N^*$ for $\{1,2,\cdots\}$

  • Positive or negative is taken in strict sense in U.S and loose sense in E.U (this means $0$ is both a negative and a positive number).

Thus $\mathbb Z^+$ or $\mathbb R^+$ in E.U contains $0$ and not in U.S, as for the case of $\mathbb N$ we can use $\mathbb Z_0^+$ and $\mathbb R_0^+$ for expliciting the set contains $0$.

In E.U we would use $\mathbb Z^{+*}$ or $\mathbb R^{+*}$ to exclude $0$.


As you can see U.S add zero via the notation $\mathbb X_0$ while E.U exclude it via $\mathbb X^*$.

  • Even with the interval notation there are differences (but only in the symbols used), U.S would generally note open bounds with parenthesis while E.U would use outward facing square brackets.

i.e. $(a,b) \text{ vs } ]a,b[$ or $[0,+\infty)\text{ vs }[0,+\infty[$

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