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?

Solution 1:

"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 $x \lt 0$.

Solution 2:

$\{ g \in G : \Phi(g) \}$ 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 Personally, I like to use $\ni$, which is a (somewhat outdated) alternative such that symbol.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 $\text{s.t}$.

Solution 3:

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.

Solution 4:

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.

Solution 5:

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.