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On my recent exam, my professor wrote a note saying that the following notation shouldn't be used when discussing domains, and I'm not sure why it matters. So, my question:

I'll use a specific example, but I'm speaking generally when it comes to notation. When given a function of the form:

$$f(x) = \cases{ g(x) & if $x=1,2,3,...$ \\ h(x) & if $ x= -1, -2, -3, ..$} $$

Is it incorrect to write it as:

$$f(x) = \cases{ g(x) & if $x\in \mathbb N$ \\ h(x) & if $ x\in \mathbb Z<0$} $$

I've always been curious why this notation isn't used in textbooks, and my professors comment on my exam peaked my curiosity enough to post it here.

EDIT: I should mention I come from a physics and engineering background, not mathematics.

Thank you!

NoVa
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    In fact the second version is the usual way to write it down. A bit better would be $x\in${$1,2,3\cdots$} , for example. – Peter Apr 24 '18 at 21:21
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    I agree with your professor for several reasons. The first is that the first way to write it is easier and faster ro read and understand than the second. The second is that there is sometimes some ambiguity about whether $0$ is included in $\mathbb{N}$. I know the convention - it's not - but don't like to rely on my reader knowing that. @Peter 's suggestion that you write the sets explicitly is OK too. – Ethan Bolker Apr 24 '18 at 21:23
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    @EthanBolker, so why couldn't you say "$\mathbb Z \ge 0$" instead of $\mathbb N$? – NoVa Apr 24 '18 at 21:24
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    Well, according to a professor of the university in Kaiserslautern, there is no convention whether or whether not $0$ belongs to the natural numbers. What about "positive numbers" ? For me this clearly rules out $0$, but it seems not everyone agrees to that. – Peter Apr 24 '18 at 21:26
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    it is strange to write $\Bbb Z<0$ but it can be understood. My usual notation is $\Bbb Z_{<0}$ or $-\Bbb N_{>0}$, what I think is enough clear. Also you can write something more formal like $x\in\Bbb Z,\land, x<0$ – Masacroso Apr 24 '18 at 21:27
  • @Masacroso, I've never seen your notation used before. – NoVa Apr 24 '18 at 21:28
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    the point on notation is that anyone can understand it. It is common, prior to use it, define it meaning, then you can use the notation that you please. – Masacroso Apr 24 '18 at 21:30
  • @Masacroso I agree, but most mathematicians would assume that "positive integers" is absolutely clear. However, some interprete it as "non-negative" and use "strictly positive" to rule out $0$. Maybe some more conventions (like that the square root is always the positive value) would be helpful to avoid confusions. – Peter Apr 24 '18 at 21:36
  • I think most readers would understand something like $\Bbb{Z}_{<0}$ but would be baffled by $\Bbb{Z} < 0$. I think the majority of readers in the $21^{\mbox{st}}$ century would find it rather quaint for $0$ not to be a member of $\Bbb{N}$. – Rob Arthan Apr 24 '18 at 21:45
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    @Peter: using "positive" to mean "non-negative" is sloppy and bad: it is good for everyone to be aware that some writers make this mistake, but we shouldn't endorse it. – Rob Arthan Apr 24 '18 at 21:49
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    @Peter "What about positive numbers? For me this clearly rules out $0$ but it seems not everyone agrees to that" I have never heard of anyone who suggests that zero is a positive number (it is not). I have heard of people who suggest that $\Bbb N$ should represent the set of positive integers, and I have also heard of people who suggest that $\Bbb N$ should represent the non-negative integers. Noone who suggests that $\Bbb N$ should include zero is doing so under a misunderstanding of the meaning of the word "positive." – JMoravitz Apr 24 '18 at 21:52
  • @kosta: is it intended that $f(x)$ is not defined for $x = 0$? – Rob Arthan Apr 24 '18 at 21:53
  • @RobArthan, for my specific example, yes. – NoVa Apr 24 '18 at 21:57
  • @JMoravitz A user on this site with a very high reputation (I do not remember the name) posted that for him/her, zero belongs to the positive integers, which baffled me. I am calmed down to hear that I am not the only one finding this strange. – Peter Apr 24 '18 at 21:59
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    @Peter: it would be good if you could find that post. "Positive" must surely mean "greater than zero" in any reasonable context. – Rob Arthan Apr 24 '18 at 22:10

1 Answers1

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Ignoring the question of zero's status w.r.t. $\mathbb{N}$: There is some flexibility and looseness in notation that most people will allow, but $$x \in \mathbb{Z} \lt 0$$ just doesn't work (for me).

Here's why: people often write $$ a R_1 b R_2 c$$ as an abbreviation of $$a R_1 b \text{ and } b R_2 c,$$ where $a$, $b$, and $c$ are variables or constants and $R_1$ and $R_2$ are binary relation. So you can write $a \lt b \lt c$ to mean $a\lt b$ and $b \lt c$, or even $0 \lt x \in \mathbb{R}$ to mean $0 \lt x$ and $x \in \mathbb{R}$, but your expression leaves me with $\mathbb{R} \lt 0$ for the second half, which is either confusing or nonsensical.

I guess you could stay within the limits I've prescribed and write it as $$ 0 \gt x \in \mathbb{N},$$ which, if I came across, I would think "Hmm, that's a little weird, never seen it before" but it would be clear to me what you meant. Math is a language and there are lots of sentences that are grammatical and you could write, but people generally don't write. So using them adds a bit of unnecessary confusion that it's probably better to avoid if you can.

Regardless, I find your first way of writing it more straight forward and would prefer it. It also lets you avoid all the hubbub about $0 \in \mathbb{N}$ versus $0 \notin \mathbb{N}$ that you see in the comments.

JonathanZ
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