I am not so sure whether the meaning of $$\mathbb Z_+$$ is very clear. How many different definitions are there? Does the definition that is used depend on whether the writer is English or German?
In French maths, this notation doesn't exist.
Asked
Active
Viewed 334 times
5
-
I think that we use the notations : $\mathbb{Z}={\ldots,-2,-1,0,1,2,\ldots}$, $\mathbb{Z}_+={0,1,2,\ldots}$ and $\mathbb{N}={1,2,\ldots}$ – Thibaut Dumont Jun 07 '13 at 13:55
-
It depends on the context. A guess is that it means the additive group of $\Bbb Z$, i.e. $\Bbb Z$ when one considers the addition only. – Andrea Mori Jun 07 '13 at 13:56
-
Without any doubt $\mathbb N$ includes $0$. – Damien L Jun 07 '13 at 13:59
-
8@DamienL, alas there is doubt. – vadim123 Jun 07 '13 at 14:00
-
1@DamienL Not true. I define $\mathbb{N} = \varnothing$, then $0\not\in\mathbb{N}$. My point is that you can't argue in general without stating your definition. BTW, there are people who define $\mathbb{N} = {1,2,\cdots}$ – mez Jun 07 '13 at 14:08
-
hoho… Thank you for pointing that out vadim ! – Damien L Jun 07 '13 at 14:15
-
5@vadim123: In fact $0\in\Bbb N$ is a counterexample to the principle of the excluded middle: it is a proposition that contains no variables, yet it is neither true nor false ;-( – Marc van Leeuwen Jun 07 '13 at 14:38
3 Answers
7
Many people would interpret this to mean $\{1,2,3,\ldots\}$, although some might argue for $\{0,1,2,3,\ldots\}$. Absent any other context I don't think any other interpretations are likely.
Sadly, many authors use notations without defining them, because they are "standard" in their little corner of mathematics.
vadim123
- 82,796
-
That would more likely be $\Bbb Z_{>0}$ or $\Bbb Z_{\geq0}$, but everything is possible! – Andrea Mori Jun 07 '13 at 13:58
-
1Some people still uses : $\mathbb{R}+=[0,\infty)$ for $\mathbb{R}{\geq 0}$. – Thibaut Dumont Jun 07 '13 at 14:05
-
1Which causes my stupefaction about how people could "resolve" the ambiguity of the symbol $\Bbb N$ with another one that raises exactly the same problem! – Marc van Leeuwen Jun 07 '13 at 14:41
5
Symbol "$+$" means "positive", so $\mathbb{Z}_+$ shoud properly be understood as $\{1,2,3,\ldots\}$. It's less confusing that undefined $\mathbb{N}$ in some paper, where we don't know if it includes $0$.
Bartek Pawlik
- 657
-
1It's interesting because in french "positif" means non-negative. So it needs context probably. – Thibaut Dumont Jun 07 '13 at 14:07
-
4Actually $+$ means addition, not positive. If $n<0$ then $+n$ is (also) negative while $-n$ is positive. Therefore $\Bbb Z_+$ is an abomination that only makes the embassement of not being able to assign a uinque truth value to $0\in\Bbb N$ even worse; it should have been $\Bbb Z_{>0}$ (or even better $\Bbb N_{>0}$), which would never have raised the current question. – Marc van Leeuwen Jun 07 '13 at 14:36
-
It depends on context. For example, when we are talking about electric fileds, positive electric charge is labaled by "$+$", and negative by "$-$" (not by $">0"$ nor $"<0"$). Thus I don't see anything wrong in labeling positive numbers by "$+$". This example could be used also as a respond to @Thibaut Dumont: I think that putting "$0$" as a positive number (like in french notation) is not too good ($0$ is not positive nor negative; it's neutral). – Bartek Pawlik Jun 07 '13 at 15:38
1
For the French notation $\mathbb N=\{0,1,2,3,\ldots\}$ and $\mathbb N^*=\{1,2,3,\ldots\}$ and and the notation $\mathbb Z_+$ is not used in general but it means $\mathbb N$.
-
In the U.S. it is far from standard but common is $\mathbb{N}={1,2,3,\ldots}$ and $\mathbb{N}_0={0,1,2,\ldots}$. – vadim123 Jun 07 '13 at 14:15