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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?

Mike Pierce
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bjd2385
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3 Answers3

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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.

Jim
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    I wouldn't restrict the usage of colon ($:$) to set-builder notation only. While $|$ can easily become ambiguous due to the absolute value, $:$ cannot and serves as a visual separator between, say, quantors and the quantified expression. For example $$\forall \varepsilon > 0 \ \exists \delta > 0 : |x-y|<\delta \Rightarrow |f(x)-f(y)| < \varepsilon\ \forall z\in \mathbb R\ \exists w\in \mathbb C : w^2 = z$$ – AlexR Feb 04 '15 at 12:05
  • True, if you want to write in symbols as you have there. But I'd say most mathematical papers would, as a matter of style, strive to avoid such a dense mess of symbols. I also think the abbreviation "s.t." is better suited for when you really need something like that in symbols. – Jim Feb 04 '15 at 18:32
  • I agree with you if any side of the s.t. gets larger (wich is usually the case in papers), but compact statements like those above can be kept readable in this compact notation as well :) – AlexR Feb 04 '15 at 18:33
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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 $\ni$.

nomen
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  • Interesting comment as I had actually never heard or seen Giuseppe Peano mentioned before. – bjd2385 Feb 03 '15 at 22:02
  • Also news to me. “According to Julio González Cabillón, Peano introduced the backwards lower-case epsilon for "such that" in Formulaire de Mathematiques vol. II, #2 (p. iv, 1898).” http://jeff560.tripod.com/set.html – Matthew Leingang Feb 04 '15 at 02:12
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    @Matthew, thanks for the pointer. In this scan the ɜ looks more like a reversed letter epsilon than a comma to me. Since ɛ is apparently used to mean “element of” in that text, rendering ɛ as $\in$ and ɜ as $\ni$ in modern typography does make sense to me. Still wouldn't use the symbol with that meaning these days. – MvG Feb 04 '15 at 11:48
  • But then note that Peano "used an epsilon for membership in Arithmetices prinicipia nova methodo exposita", same page in "earliest uses". According to this, the backwards $\in$ is a backwards $\in$ since it is a backwards ϵ and $\in$ is a normal ϵ. – A. Donda Feb 04 '15 at 17:51
  • @MvG: I am editing my answer slightly to separate Peano's usage from the modern (if rare) usage of backwards epsilon for s.t. – nomen Feb 04 '15 at 19:31
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The problem I have with $\ni$ is that it makes an optical illusion in combination with $\in$: $$ \forall z \in \mathbb{R} \ \exists w \in \mathbb{C} \ni w^2 = z $$ My eye is drawn to the $\mathbb{C}$ and I can't focus on the formula. Also there's the issue that $A \ni x$ is used as syntactic sugar for $x \in A$, and I find this more useful.

In written work I would use no symbol for “such that”—just the words. In notes I will use “s.t.” which works fine. $$ \forall z \in \mathbb{R} \ \exists w \in \mathbb{C} \text{ s.t. } w^2 = z $$ When logicians use $\exists$, the “such that” is implicit. So you could technically use no symbol, but I don't know how readable that is.
$$ \forall z \in \mathbb{R} \ \exists w \in \mathbb{C} (w^2 = z) $$ Actually I think they write it more like this: $$ \forall z (z\in\mathbb{R} \rightarrow\exists w(w\in\mathbb{C} \wedge w^2=z)) $$

Matthew Leingang
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    I like your explanation, it makes practical sense, i.e. $\ni$ would draw unnecessary attention to certain parts in combination with the former $\in$. – bjd2385 Feb 03 '15 at 20:11
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    I think that, just as $\forall x$ means "for all $x$", $\exists x$ means "for some $x$" or equivalently "there exists $x$ such that", so there is no need for "such that" after $\exists$. – bof Feb 04 '15 at 06:20
  • @bof well put. Totally stealing that. – Matthew Leingang Feb 04 '15 at 09:34
  • I usually write a $:$ after $\exists$ to be read as "such that" while preventing that I have to add parentheses around the entire statement: $$\forall \varepsilon > 0 \ \exists \delta > 0 : |x-y|<\delta \Rightarrow |f(x)-f(y)| < \varepsilon\ \forall z\in \mathbb R \exists w\in \mathbb C : w^2 = z$$ – AlexR Feb 04 '15 at 11:59