# “Such that” symbol “$\mid$”

I know the "such that" symbol $\mid$ from the definition of sets: $$\{x \mid x \in \Bbb N \land x < 3\}$$

Is it OK to use this symbol outside of sets. For instance, if I want to define a function that takes a non-empty set of natural numbers and yields the least element of this set, can I write:

$$f : \mathcal P (\Bbb N) \setminus \{ \emptyset\} \to \Bbb N \\ x \mapsto y \mid y \in x \land \forall z: z \in x \to z \geq y$$

Or would a mathematician shoot me on sight, if I wrote this?

EDIT:

Thank you for your comment. One proposition you made was to write "such that" in words. But doesn't this break the goal of a formal notation, i.e. its international comprehension. If I wrote: $$x \mapsto y \text{ tal que } y \in x \land \forall z: z \in x \to z \geq y$$ or $$x \mapsto y \text{ tal que } y \text{ sea el elemento mínimo del conjunto } x$$

Wouldn't this lead to misunderstandings if the reader didn't speak Spanish?

To make the question short: How would you write down the function $f$ as defined above?

• Mathematicians are generally unarmed. – Gerry Myerson Mar 2 '13 at 5:46
• @GerryMyerson: Generally unarmed $\neq$ unarmed – JavaMan Mar 2 '13 at 5:47
• I am a strong proponent for using the words "such that" or at least "s.t." as it is not much more work that writing $\mid$ and much more clear. – JavaMan Mar 2 '13 at 5:48
• I'm not sure that the goal of formal notation is "international comprehension". Besides, it's difficult to have a substantive conversation about mathematics without employing words as well as notation. – PersonX Mar 2 '13 at 6:21
• In strong agreement with @Chris, I would like to point out that most published mathematics is mostly words, not mostly symbols. – Lubin Mar 2 '13 at 6:45

## 3 Answers

You ask how I would write this function:

$$f : \mathcal P (\Bbb N) \setminus \{\} \to \Bbb N \\ x \mapsto y \mid y \in x \land \forall z: z \in x \to z \geq y$$

First I’d correct the error in the top line: you want the domain to be the family of non-empty subsets of $\Bbb N$, which is $\wp(\Bbb N)\setminus\{\varnothing\}$ or, if you insist on avoiding the standard notation for the empty set, $\wp(\Bbb N)\setminus\{\{\}\}$. Your $\wp(\Bbb N)\setminus\{\}=\wp(\Bbb N)\setminus\varnothing=\wp(\Bbb N)$. The rest is easily compressed into one line:

$$f:\wp(\Bbb N)\setminus\{\varnothing\}\to\Bbb N:x\mapsto\min x\;.$$

In my view $y=\min x$ is much easier to grasp than ‘$y$ is the unique element of $x$ such that $y\le z$ for all $z\in x$’, whether the latter is expressed in English, in Spanish, or entirely in mathematical symbols.

For the more general question, I would no more use $\mid$ for such that in general than I would use the colon that I prefer for my set notation: I would not expect it to be automatically understood (and would not immediately understand it myself). In the given context I would understand tal que immediately, and my Spanish is very, very minimal.

I don’t think consider international comprehensibility to be a major goal of mathematical notation, formal or (relatively) informal. The primary function of good mathematical notation in everyday mathematical use is to make the mathematics easier to understand and follow. (Notation intended to aid mechanical theorem-proving or the like is an exception.)

• Thank you very much. I corrected the domain. Your definition of $f$ looks very nice and understandable to me. I wasn't aware that "min" was in common use as a function over sets. – Hyperboreus Mar 2 '13 at 16:42
• @Hyperboreus: You’re very welcome. Yes, $\min$, $\max$, $\inf$, and $\sup$ are all in common use in such contexts. – Brian M. Scott Mar 2 '13 at 16:47

I would go a bit farther even than Brian's answer, and use more words and fewer symbols. (I'm not sure that his second colon is standard, by the way.)

I would simply say "given a nonempty set $A$ of natural numbers, we denote its least element by $f(A)$." As Brian says, this particular function $f$ is often simply called $\min$.

• I believe it is standard in the form $f\colon x\mapsto f(x)$ (myself, I would even write $f=(x\mapsto f(x))$). I'm not sure if I've seen two colons used in a row (one to specify the domain and codomain and the other to specify the function), but I think it's rather clear. – tomasz Mar 2 '13 at 17:07
• @tomasz Yes, I'm fine with $g:A \to B$ and $g:a \mapsto b$. I just haven't seen $g:A \to B:a \mapsto b$. – Trevor Wilson Mar 2 '13 at 17:10
• @tomasz The statement $f=(x \mapsto f(x))$ is confusing to me because I think "$x \mapsto y$" is a statement and not an object. Perhaps $f = \lambda x.f(x)$ would be better? – Trevor Wilson Mar 2 '13 at 17:13
• I don't think all mathematicians are familiar with lambda calculus, whereas I'm pretty sure everyone is (or should be!) familiar with $\mapsto$. ;) I'm not saying that equality is better than a colon. The latter is more standard and the only one I would use in very formal context, I suppose. I just find it rather intuitive to say it that way (with $=$). – tomasz Mar 2 '13 at 17:16
• @tomasz Yes, I wasn't actually advocating the use of lambda calculus notation (I prefer to reserve "$\lambda$" for ordinals anyway.) Although many mathematicians are familiar with "$\mapsto$", not as many can parse the statement "$f = (x\mapsto f(x))$". I know I can't. – Trevor Wilson Mar 2 '13 at 17:20

At the university I attended, s.t. and .э. (some profs omitted the periods on the later case) were used. However, I doubt these are universal and will mimic others when saying written language is usually the best way to convey meaning. Coming from a statistics background, for example, | typically means "given" to me rather than "such that" like some people use in set notation (I use a colon as "such that" in set notation).

Also, things like э can be used as, or at least look familiar to, "belongs to" notations for sets. Mathematicians generally use, recycle, and then reuse notations again and again, so in a situation where you're speaking to a general audience-- words are your best friend.