Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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^{-+} $

Thanks in advance!

share|improve this question
6  
I would use the second one. I have never seen the first notation before, to be honest :) –  Avitus Jun 28 '13 at 18:52
7  
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

4 Answers 4

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.

share|improve this answer
    
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.

share|improve this answer
6  
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.

share|improve this answer
4  
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

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 $\mathbb{Z} \setminus \{0\}$.

share|improve this answer
    
Or just "nonzero integers". –  Thomas Jun 29 '13 at 1:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.