# A quickie about set theory notation

I'm reading the first chapters of my discrete mathematics textbook and I couldn't help but wonder (perhaps I haven't seen enough examples) -- is it more appropriate to write that $a$ is an integer and within a set in the following way: \begin{align} a\in\left\{x\in\mathbb{Z}\big|\:b<a<c\right\},\tag{1} \end{align} or would it be better to break it up into two statements instead to have something like \begin{align} a\in\mathbb{Z},a\in\left(b,c\right)\tag{2}. \end{align} And I apologize in advance if some of you feel this is not a good question (perhaps too beginner-ish). I would just like to get all of the formalities down so that in the future I don't make a mistake.

Thanks,

• I prefer (2) to (1). Another option is $a \in \mathbb{Z} \cap (b,c)$, which may be useful in some contexts but perhaps too formal in others. A further possibility is $[[ b + 1, c- 1]]$. That's not exactly right; it's supposed to be \llbracket and \rrbracket. Jan 20, 2015 at 3:44
• Thanks, that makes sense actually. Great suggestion. +1 Jan 20, 2015 at 3:45
• One could simply speak of "integral $a$ strictly between $b$ and $c$". There's nothing wrong with legible mathematics.
– MPW
Jan 20, 2015 at 3:54
• For nitpicking completeness: This question says $\;a\in\left\{x\in\mathbb{Z}\big|\:b<a<c\right\}\;$ but the intention is of course $\;a\in\left\{{\boxed a}\in\mathbb{Z}\big|\:b<a<c\right\}\;$ or $\;a\in\left\{x\in\mathbb{Z}\big|\:b<{\boxed x}<c\right\}\;$. Mar 7, 2018 at 20:21

My personal preference is to write something like: "let $a$ be an integer, where $b < a < c$." There's nothing wrong with mixing in statements in natural language.
• If you wanted to go in the spirit of the latter you could do $a \in \mathbb Z \cap (b,c)$. But as GFauxPas said, it's more important that you're clear. Jan 20, 2015 at 3:44