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?
Best Answer
In modern mathematical papers $\ni$ is almost exclusively used to mean contains, as in, "the integers contain $3$" would be written as $\mathbb Z \ni 3$. If you intend that others read your mathematics then I would highly recommend you stick to using $:$ or $|$ for "such that" in set builder notation. If your notation is just for your own use then it doesn't really matter what you use as long as you know what it means.