# 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?

• 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 $x \lt 0$.

• 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. – Namaste Feb 20 '13 at 21:48
• 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
• @amWhy: Sorry for trashing your comments section like that! :-) – Asaf Karagila Feb 20 '13 at 22:24

$\{ g \in G : \Phi(g) \}$ 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 $\text{s.t}$.

• Exemplify the answer of the problem is the best way(+1) – mrs 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"). – user26486 Sep 2 '14 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 '14 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. – user26486 Sep 3 '14 at 5:51
• @mathh Very interesting. Which country is that? – Alexander Gruber Sep 3 '14 at 13:37

The symbol used in my experience is not the ordinary backwards epsilon but is similar to ⋺ Unicode 22FA (hex). As some have noted, it also differs by the length of the median bar which penetrates the curved outer shell.

Since I'm trying to write in a form that facilitates translation of notes into Notes – IE Semantic Normal Form – I use many symbols, and am trying to learn more.

My experiments attempt to include logics, more discussed by philosophic logicians, to clarify contexts and contingencies like mood, time relations, belief states, type of evidence, modality of argument, and the like. I would welcome references to others learning and playing with similar topics.

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.

• 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

As people have stated, you can write "$a\in\{b|p_1(b)\}$", but even better just write "$p_1(a)$".

eg. If your property $p_1(b)$ is $\forall x\in y,\ b\in x$ and you want to say $a$ is such a thing you can just write $\forall x\in Y,\ a\in x$. No need to write $a\in\{b|\forall x\in Y,\ b\in x\}$.

This is much more concise, easier to understand, and less cumbersome in notation.

Also, in axiomatic set theoretic terms, after all, formulas (of which "$\forall x\in Y,\ b\in x$" is one) are primary, and you don't necessarily know that $\{b|p_1(b)\}$ is a set, in general.

• I didn't understand your last two lines. We just need set-notation after all. – Daniel Buck Aug 15 '18 at 20:52
• Put it this way: it's better to say "I am a man" than to say "I belong to the set of things that are men". What I meant in the last two lines is that in the formal first-order language of predicate logic, $\{x|\phi\}$ is not part of the language but stands in for a set of abbreviations; $y\in\{x|\phi\}$ is technically an abbreviation for $\phi(y/x)$ (the formula $\phi$ but replaces instances of $x$ by $y$), and some strings of the form $\{b|p_1(b)\}$ do not define sets. Everybody knows Russel's Paradox. – toe-pose Aug 16 '18 at 15:03

Another similar "such that" symbol is $\ni'$. e.g. $$a\in \left\{A\ni' a > 0\right\}.$$Using $\ni$ alone can cause quite some confusion, so adding a small dash $'$ indicates that this is not a symbol to denote membership/containment; rather, it denotes "such that." Also, there are actually two commands to grenerate $\ni$, namely, \ni and \owns (explanation for the latter command can be found in an answer to this post).

There is also the symbol $\stackrel{\bullet}{\equiv}$ to denote "such that" which is very uncommon, but I sometimes like to use it, though I never use it when posting questions or answers here as I assume many users will not know what it means. e.g. $$\exists x \stackrel{\bullet}{\equiv} x\in X.$$ There is not a nice command to typeset this symbol, either. It is \stackrel{\bullet}{\equiv}. I only knew about this after looking at a PDF that I do not remember now, and it used this notation to denote "such that."

However, if you would like to use it, type \newcommand{\...}{\stackrel{\bullet}{equiv}} and whatever the command \ $\ldots$ is, that is now your new (and hopefully easier) command to create this symbol. I would use \st since it stands for such that, but you can use any new command you like.

Also, I have seen $\$$be used to denote such that (because it kind of looks like$S$and$T$put together, which is kinda cool), but if you ever need to user super-factorials, I do not suggest this notation. A better, but less common, notation is$\varepsilon$(formatted with $\varepsilon$). You can find answers and comments related to that latter symbol$\varepsilon$in the linked posts. I know this question is over$5\$ years old, but for people who search on Google, "what does 'such that' mean?", this post is one of the first links to come up. So, for their sake, I have decided to share this answer.