# Is this proper notation: $\;\mathbb Z^{+-}\;$?

I have a question about notation. Is it okay to represent $\;\mathbb Z^{+-}$ as $\mathbb Z - \{0\}$ ?

Like does it make sense to manipulate both sets mathematically like that? ) i.e. $\{0\} = \mathbb Z - \mathbb Z^{-+}$

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I would use the second one. I have never seen the first notation before, to be honest :) –  Avitus Jun 28 '13 at 18:52
Using the minus for set difference (i.e. $X - Y$ as an alternative notation for $X \setminus Y$) is widespread. Unless you mix it with $A - B = \{ a - b \colon a \in A, b \in B \}$, there's no problem. –  Daniel Fischer Jun 28 '13 at 18:52
thanks for the prompt reply fellas. have a good one. –  tuba09 Jun 28 '13 at 18:52
sorry about that. didn't mean anything by it. –  tuba09 Jun 28 '13 at 18:56

Using $\mathbb Z^{+-}$ would have given me little clue as to what you are denoting, were it not for the equality you use to define it. (Okay, I could have probably guessed, but why make your readers have to guess?)

I would strictly use $\mathbb Z - \{0\}$, or better yet, $\mathbb Z \setminus \{0\}$. I'd even prefer it were spelled out in words: "all non-zero integers" than trying to guess what your notation denotes. (I have never seen the notation $\mathbb Z^{+-},\,$ to be honest.)

I don't think it makes sense to write $\{0\}$ as $\mathbb Z - \mathbb Z^{-+}$. (I'm not clear why you'd want to avoid using $\{0\})$. In any case, using $\;\{0\}$ is much more straightforward, and easier to write and read.

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noted for future reference! –  tuba09 Jun 28 '13 at 19:00
Always good to ask, when in doubt! :-) –  amWhy Jun 28 '13 at 19:01

The notation $\Bbb Z^{+-}$ is confusing. I'd probably write $\Bbb Z_0$ for $\Bbb Z\setminus\{0\}$. While there are several meanings for the notation for $\Bbb Z_n$ when $n\neq 0$, there are none (that I am aware of) in the case of $n=0$.

Of course you can, and should, always define clearly new symbols that you are using.

As Rahul suggests in the comments, a much better idea is to use $\Bbb Z_{\neq0}$ or $\Bbb Z^{\neq0}$, this instantly tells you what is the content of the set.

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I'd suggest $\mathbb Z_{\ne0}$, which makes it totally clear. I always prefer when people write $\mathbb R_{\ge0}$ or $\mathbb R_{>0}$ instead of $\mathbb R^+$. –  Rahul Jun 28 '13 at 19:09
@Rahul: I totally agree. I'm going to steal this idea and add it into my answer! –  Asaf Karagila Jun 28 '13 at 19:11
I've seen those superscripted as well, sometims. @RahulNarain Such as: $\mathbb R^{\geq 0}$. In algebra with fields, of course, $\mathbb k^*$ is the non-zero elements of $k$, but that doesn't work with non-fields. –  Thomas Andrews Jun 28 '13 at 19:11
@Thomas: I actually had an instinct to write $\Bbb Z^\times$, but that one is the units, which in a field means non-zero elements, but in $\Bbb Z$ means only $\pm1$... –  Asaf Karagila Jun 28 '13 at 19:24

$\Bbb{Z}-\{0\}$ is usually denoted by $\Bbb{Z}^*$, the set of invertible integers.

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Only $\pm1$ are invertible. –  Asaf Karagila Jun 28 '13 at 19:24
I was confusing $\mathbb Z^{-+}$ with $\mathbb Z^*$ that's the one that the math department at my university uses. –  tuba09 Jun 28 '13 at 19:26
In France,$\mathbb{Z}^*$ sometimes means $\mathbb{Z} - \{0\}$, and sometimes $\{+1, -1\}$, we never know. –  justt Jun 28 '13 at 19:29
@justt: One more reason to study probability well. You can calculate the probability of it being one or the other! Then you can say that a certain statement is true in probability $p$. –  Asaf Karagila Jun 28 '13 at 19:32
@AsafKaragila I'm referring to $\Bbb{Z}$ as a subring of $\Bbb{R}$ so invertible means invertible in $\Bbb{R}$ –  metacompactness Jun 28 '13 at 20:24
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For your own purposes you can use any notation, but it probably wouldn't be understood by others without qualification. $\mathbb{Z}^*$ stands for the set of invertible integers, while $\mathbb{Z}^{\times}$ means integers except zero. Personally I'd simply write $\mathbb{Z}\setminus 0$, or if you prefer more rigorously $\mathbb{Z} \setminus \{0\}$.

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Or just "nonzero integers". –  Thomas Jun 29 '13 at 1:27