# Are the following sets exactly the same: $\mathbb{N}$, $\mathbb{N_{-\{0\}}}$, $\mathbb{N^+}$ and $\mathbb{Z^+}$?

At university, my Maths Analysis lecturer said that this is still debated nowadays. For example, a definition of an rational number is $\cfrac{p}{q}$ where $p\in \mathbb{Z}$ and $q\in \mathbb{N}$ (or should it be $q\in \mathbb{N^+}$ or the other two in the title?).

I know that $\mid \mathbb{N}|= \mid \mathbb{N^+}|$ since I can find a bijection by defining the map: $f: \mathbb{N^+} \mapsto \mathbb{N}$ by $f(x)=x-1$.

But the problem is that this raises the question as to whether $0 \in \mathbb{N}$ or $0 \in \mathbb{Z^+}$. This is similar to asking whether $0$ is positive?

So which set should I use to define the denominator $q$ of a rational number and why? The choices are $\mathbb{N}$, $\mathbb{N_{-\{0\}}}$, $\mathbb{N^+}$ and $\mathbb{Z^+}$.

I acknowledge that this question is subject somewhat to opinion, but I value the opinions of users of this site. So all I am asking is which set of the above would you choose $q$ to belong to for the definition of the rational number?

• In class-related work you should use what your lecturer uses. Elsewhere, you can use what you like and is most convenient for what you want to do. Maybe you work closely with other people that already have a standard. If the distinction is important for what you do, you should mention whether or not $0 \in \mathbb{N}$. Dec 8 '15 at 23:21
• (1) We don't need a tag for "mappings" because those are in fact functions. (2) This question is about notation, and not at all about functions, so the tag (and the functions tags) is entirely irrelevant. Dec 10 '15 at 10:38
• @AsafKaragila Okay, understood; thanks for letting me know. Dec 10 '15 at 10:41

There are two different sets of relevance here; each has several notations in use.

The set $\{0,1,2,3,\ldots\}$ is considered fundamental in areas such as logic, set theory and computer science. It can be unambiguously written $\mathbb N_0$. Set theorists often refer to it as $\omega$, deftly avoiding the notational trouble surrounding the $\mathbb N$s.

The set $\{1,2,3,4,\ldots\}$ is often needed in most of the rest of mathematics, where people find it more natural to start counting at $1$. It can be unambigously written $\mathbb N_+$ or $\mathbb Z_+$ (the location of the plus sign can vary).

Either of these sets is often notated just $\mathbb N$ -- it is up to the reader who encounters this to know (or guess) which convention the author is following, in the cases where the difference matters. It is considered polite for an author to state which convention he follows before he uses the naked $\mathbb N$, but this is not always done in practice.

In English it is unambiguous that $0$ is not "positive". However other languages may follow other conventions; in French the number $0$ counts as both "positif" and "négatif" and one has to speak about "strictement positif" if one needs to exclude $0$. (This is not a mathematical difference; the concepts $>0$ and $\ge 0$ both exist independently of which words we use about them, and the two languages simply chose different concepts to have a short word for).

• Nice answer, thanks very much. So in your opinion, which of the four would you select, I know it's subject to opinion and therefore arbitrary, but could you tell me which one you would use? Dec 10 '15 at 7:59
• @BLAZE: Personally? I just use $\mathbb N$ for whatever of the sets I'm speaking about (usually, but not always, the one that contains $0$) unless there seems to be a particular risk that the reader will miss my point if he misunderstands me. In that case I write $\mathbb N_0$ or $\mathbb N_+$. (Except forsuch set-theory contexts where it is conventional to use $\omega$). I'm not holding this up as a shining example to follow, though. Dec 10 '15 at 15:44

It usually depends on the course and the lecturer.

$0 \in \Bbb N$ vs $0 \not \in \Bbb N$ depends on the definition of $\Bbb N$. The first one is usually found in most logic courses, while on say, analysis, this is not as common.

Whenever I see $\Bbb Z^+$ I'll think they refer to $\{1,2,3,...\}$, same for $\Bbb N^+$(this last one is quite uncommon).