11

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?

Trevor Wilson
  • 16,989
Hyperboreus
  • 341
  • 1
    I don't think it is a common use. –  Mar 02 '13 at 05:42
  • 18
    Mathematicians are generally unarmed. – Gerry Myerson Mar 02 '13 at 05:46
  • 2
    It is more likely that you would get frowned at instead of shot. – treble Mar 02 '13 at 05:47
  • 6
    @GerryMyerson: Generally unarmed $\neq$ unarmed – JavaMan Mar 02 '13 at 05:47
  • 13
    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 02 '13 at 05:48
  • Thank you for your comments. Please see my edit. – Hyperboreus Mar 02 '13 at 06:10
  • 4
    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 02 '13 at 06:21
  • 2
    In strong agreement with @Chris, I would like to point out that most published mathematics is mostly words, not mostly symbols. – Lubin Mar 02 '13 at 06:45
  • 1
    On our team, the nationals speak very bad English and the US-Americans that work with us, even worse Spanish. Between the developpers/coders, actually a lot of communication is done by skribbling code snippets on the walls and -true- doing the rest in very basic english/spanish. Now our analysts are more of the mathematical persuasion and I thought, that mathematical notation could be almost as international as code snippets. But obviously, this doesn't hold. – Hyperboreus Mar 02 '13 at 06:48
  • For the record, the only formal notation for "an $x$ such that $P(x)$" that I've ever seen is that used in Hilbert's epsilon calculus. Using that notation, the definition of $f$ would be written as $x\mapsto \varepsilon y(y\in x \land (\forall z\in x)(y\leq z))$. – jwodder Mar 02 '13 at 06:49
  • Its quite commom to professional mathematicians to know 3 or 4 languages, at least a minimum to read papers from another country. I live in Brazil, and here you must at least know 3 languages to get a doctor title. I think thats a way to surpass this "gap". – Integral Mar 02 '13 at 14:31
  • A symbolic form of "such that" which is used in sentences is $\cdot!\ni!\cdot$. – Cheerful Parsnip Mar 02 '13 at 17:10
  • 1
    @Jim I read that once also. However, I don't see the point of it because in predicate logic no symbol is needed: "there is an $x$ with property $P$" is simply written "$\exists x,P(x)$", and outside of the context of formal logic I think it's much better to use words. – Trevor Wilson Mar 02 '13 at 17:16
  • @TrevorWilson: I agree it is redundant, but can be useful as an abbreviation when writing less formal sentences. – Cheerful Parsnip Mar 02 '13 at 17:24
  • 1
    @Jim It can also be confused with "$x \ni y$" meaning "$y \in x$" so in my opinion "s.t." is best when one is very low on ink. – Trevor Wilson Mar 02 '13 at 23:07
  • @TrevorWilson: I won't force you to use it. Don't worry. I'm just providing information. – Cheerful Parsnip Mar 03 '13 at 02:29
  • In certain mathematics derived contexts, like the input languages of several graphing calulators, the suggested notation already has a meaning of "where". It dervives from the vertical bar subscript notations utilized for restriction and evaluated at. E.g: $f(x)|_{x=4}$ – Kevin Cathcart Oct 21 '13 at 21:25
  • @GerryMyerson Yep, we're 'armless – Peter Woolfitt Oct 06 '15 at 03:30

3 Answers3

17

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.)

Brian M. Scott
  • 616,228
  • 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 02 '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 02 '13 at 16:47
5

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 $\min$.

Trevor Wilson
  • 16,989
  • 1
    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 02 '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 02 '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 02 '13 at 17:13
  • 2
    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 02 '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 02 '13 at 17:20
  • @tomasz Another beginner question: $f = \lambda x.f(x)$... Why does this have parenthesis around the $x$ and not around the application, like $f = \lambda x.(f x)$? – Hyperboreus Mar 02 '13 at 19:23
  • @Hyperboreus: Why are you asking this of me? I'm not the one who wrote it in the first place. :) I think it's a matter of custom, if not completely arbitrary. Then again, I pretty much only heard about lambda calculus. – tomasz Mar 02 '13 at 23:01
  • @Hyperboreus I don't know either. Maybe I'm using it wrong---I have never used lambda calculus myself and I was just going by memory. – Trevor Wilson Mar 02 '13 at 23:04
2

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.

Zach D.
  • 121