TITLE: A pedantic question about notation -- "such that" symbols
QUESTION [12 upvotes]: I've seen a few different symbols for "such that" in my studies, including these four:
$$
  \ni \quad\quad \:\cdot\ni\cdot\: \quad\quad \colon \quad\quad \mid 
$$
I'm aware that either of the last two are acceptable in set notations, e.g. $\left\{x:x\in\mathbb{Z}\right\}$ or $\left\{x\left.\right|x\in\mathbb{Z}\right\}$, but I am not sure about the former two, i.e. "$\ni$" vs. "$\cdot\ni\cdot$". My discrete math textbook, Discrete Mathematics with Appplications by Epp, says the latter is "such that" whereas here on Mathematics SE I've only seen the former, and even then only rarely.
When is it appropriate to use these two "such that" symbols and when isn't it?

REPLY [14 votes]: The backwards $\in$ you are talking about isn't actually a backwards $\in$.  It is backwards epsilon.  The usage was introduced by Peano specifically to mean "such that".  
The only modern usage I've ever really seen is more like a comma (small, shallow concave-left curve, lowered with respect to the line of text) with a dash through it.  It doesn't look like a backwards $\in$ at all.  Certainly, Peano's backwards epsilons look like backwards epsilons.  But not much like the modern, highly stylized epsilon $\in$ or $\ni$.