# Symbol for “such that” (not in set)

If $A$ is a set, we can use the set notation

$$A= \{ b \mid\text{property p_1 of b}\}$$

But say $A$ is an element like $b$,

$$A = b \mid \text{property p_1 of b}$$

is this a usual notation? I am trying to say that $A$ is a $b$ that such that( $\mid$ ) it satisfies property $p_1$ of $b$, and assume that exactly one $b$ satisfies property $p_1$.

Otherwise, is there a more usual convention to express this?

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Usually you would just say that $A$ possesses property $p_1$, or that $p_1(A)$ holds. –  MJD Feb 20 '13 at 21:31
You could replace the $=$ in the first equation by an $\in$ to make $A$ an element instead of a set. –  TMM Feb 20 '13 at 21:32
The usual notation is "such that". Also note that if one writes "let $A$ be a foo such that bar" then foo should be predicative and not a variable, i.e. please don't write "let $A$ be a $b$ such that $p_1(b)$", instead write e.g. "let $A$ be a positive integer such that $p_1(A)$". –  Hagen von Eitzen Feb 20 '13 at 21:32
The point being that $b$ is completely unnecessary in the second form. You could write that simply as "Assume $A$ is s.t. $p_1(A)$." –  Thomas Andrews Feb 20 '13 at 21:56

"Such that" is occasionally denoted by \ni = $\,\ni\,$, e.g., in lecture, to save time, as a shortcut. Others, when writing in lectures or taking notes, and again, to save time, use "s.t.".

But in writing anything to submit (homework, publication), when possible, it is best to just write the words "such that".

In sets though, like set-builder notation, both $\mid$ and $:$ are used:

$$\{x \in \mathbb R \mid x < 0\}$$ $$\{x \in \mathbb R : x \lt 0\}$$

"The set of all $x \in \mathbb R$ such that $x \lt 0$.

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This notation $\ni$ for "such that" was introduced by Peano, see here. –  Math Gems Feb 20 '13 at 21:45
$\ni$ usually stands for $\in$ when you are writing in Hebrew and can't foresee the amount of spacing you will need for writing left-to-right in mid-text. :-) –  Asaf Karagila Feb 20 '13 at 21:46
Yes, I think you're correct, @Math Gems! Peano it was. –  amWhy Feb 20 '13 at 21:48
+1 for simply writing "such that". –  Douglas S. Stones Feb 20 '13 at 21:56
Of course, the very text label, "\ni", indicates the alternative meaning for $\ni$, name, $X\ni x$ being a synonym for $x\in X$. That duplicate meaning is one reason it is not used much as "s.t." –  Thomas Andrews Feb 20 '13 at 21:58

$\{ g \in G : \Phi(g) \}$ is the set of those $g$ in $G$ if $\Phi$ is true. I also see $:$ for such that in piecewise functions a lot, like $$f(x)=\left\{\begin{array}{lcl}1&:&a\in B \\ 2 &:& a \notin B\end{array}\right.$$ which reads the same way. $\{g | g \in G\}$ first gives the form of stuff that you want, then "such that" g is in wherever.

So, grammatically it seems like what you say would make sense. I have never seen it used like that though. Personally, I like to use $\ni$, which is a (somewhat outdated) alternative such that symbol. (Actually this is not exactly how it's written, as a backwards $\in$. It should be thinner and taller, like a longbow. I can't find a typesetting which works on MSE's TeX though.) The modern way to do it is to use either $|$ or $:$ in sets and mathematical expressions, but just write it out if you're anywhere else. If you must abbreviate it, write $\text{s.t}$.

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Exemplify the answer of the problem is the best way(+1) –  Babak S. Jul 5 '13 at 6:39
$f(x)=\left\{\begin{array}{lcl}1&:&a\in B \\ 2 &:& a \notin B\end{array}\right.$ can mean $f(x)=\left\{\begin{array}{lcl}\frac{1}{a}\in B \\ \frac{2}{a} \notin B\end{array}\right.$ Also, $\{g | g \in G\}$ can mean that the (always true if $g\in\mathbb Z_{\neq 0}$) statement $g | g$ belongs to the set $G$. Even the $\ni$ symbol can be ambiguous at times (meaning "contains"). –  mathh Sep 2 at 19:59
@mathh I've never seen $1:a$ meaning $\frac{1}{a}$. Can you show me an example where that is used in the literature? –  Alexander Gruber Sep 3 at 2:16
@AlexanderGruber See here. "In some non-English-speaking cultures, "a divided by b" is written a : b." I have never seen "a ÷ b" being used in my country and have only seen "a : b" instead. –  mathh Sep 3 at 5:51
@mathh Very interesting. Which country is that? –  Alexander Gruber Sep 3 at 13:37

I had actually asked my prof about this a couple weeks ago... the symbol he gave is $\ni$. So, for an existential quantifier, we have:

$$\exists \,\,x\in\mathbb{R}\ni x^2 =x$$

He said we wouldn't use it in the class, as he thought it looked not so great...

This can also be seen here: http://www.physicsforums.com/showthread.php?t=195398

I, personally, like just abbreviating it "s.t." in my notes, as it's shorter, but more clear.

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That backwards epsilon notation looks terrible. I learned it a while ago and always thought it is ugly. It would be nice if it died off. Writing "s.t." is pretty simple as an alternative. –  KCd Feb 20 '13 at 21:39
I agree that in writing mathematical English one can use $\ni$ or 's.t.' but in the case you cite, I would prefer $$\left(\exists \,\,x\in\mathbb{R}\right) \left[x^2 =x\right]$$ exactly as in set notation or predicate calculus. The such that is built into the quantifier. –  Barbara Osofsky Feb 20 '13 at 21:55