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^{-+} $
Thanks in advance!
 A: 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.
A: 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 $\mathbb{Z} \setminus \{0\}$.
A: 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.
A: $\Bbb{Z}-\{0\}$ is usually denoted by $\Bbb{Z}^*$, the set of invertible integers.
