20
$\begingroup$

I'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians?

My idea is:

$|A| > 0$

However, using the emptyset $\emptyset$ might be more intuitive:

$A\ != \emptyset$

But, I know "!=" is not permissible in math community (only in programmers).

Update

Sorry, I fixed the second equation: $|A|\ != \emptyset \rightarrow A\ != \emptyset$

$\endgroup$
  • 8
    $\begingroup$ Note that $\emptyset$ is a set, and $|A|$ is a cardinality and is not (under common interpretations) itself a set. I would use $|A|\neq 0$ or $A\neq \emptyset$ but not $|A|\neq \emptyset$. (okay...if you want to define natural numbers as appears here technically $0=\emptyset$ so my objection is moot) $\endgroup$ – JMoravitz Jun 20 '16 at 5:54
  • 37
    $\begingroup$ Actually, there is usually nothing wrong with writing "$A$ is not empty" in plain English (or whatever other language) in your text $\endgroup$ – Hagen von Eitzen Jun 20 '16 at 10:13
  • 4
    $\begingroup$ In some contexts $|A|$ means the measure of the set $A$, not the cardinality. Therefore $A\neq\emptyset$ is more universally valid notation than $|A|>0$. That said, plain written language is often the best choice for indicating non-emptiness. $\endgroup$ – Joonas Ilmavirta Jun 20 '16 at 15:07
  • 3
    $\begingroup$ I would go with the simplest way to express it and the most commonly known notation, just $A$ $\neq \emptyset $. $\endgroup$ – Iff Jun 20 '16 at 16:59
  • 33
    $\begingroup$ Did you just forget about the $\neq$ symbol since last year? $\endgroup$ – user2357112 Jun 20 '16 at 17:59
82
$\begingroup$

It is perfectly fine to write $|A|>0$. However, the simplest and most common way to write this in symbols would be $$A\neq\emptyset.$$ Note that you don't want to write $|A|\neq \emptyset$, as it is $A$ itself which you are saying is not the empty set, rather than the cardinality of $A$.

(The standard symbol in mathematics for "not equal" is $\neq$, rather than $!{=}$. You can make this symbol in $\LaTeX$ with the command \neq.)

As mentioned in user21820's nice answer below, though, it is also very common to just write this in words ("$A$ is not empty" or "$A$ is nonempty") instead of symbols.

$\endgroup$
  • 9
    $\begingroup$ +1 for several reasons, one explicitly pointing out that mathematics has it's own version of "!=" and not just implicitly using it. $\endgroup$ – pjs36 Jun 20 '16 at 6:14
  • 28
    $\begingroup$ Well != was an ASCII approximation of $\ne$ created for C, not something for which mathematics had to create "its own version of". In Wolfram Mathematica there's a "not identical" operator =!=, which looks even more similar. $\endgroup$ – Ruslan Jun 20 '16 at 7:56
  • 10
    $\begingroup$ @Ruslan Some languages use a different ASCII approximation that is /= or =/= which is even more similar to the mathematical version. $\endgroup$ – Bakuriu Jun 20 '16 at 11:12
  • 3
    $\begingroup$ @user21820: What if != was created first, and then the logical negation ! by "reverse engineering"? What if ! was used because it looks like the vertical strikes in symbols such as $\neq$ and $\not\subset$? What if? $\endgroup$ – timur Jun 20 '16 at 15:37
  • 3
    $\begingroup$ @Soke: Yes. We say that $|A|>|B|$ if there is an injection from $B$ to $A$, but no bijection between them. It's trivial to see that every infinite cardinality is greater than 0. So $|A|>0$ is a valid way to say that $A\neq\varnothing$, even if $A$ can be infinite. $\endgroup$ – Meni Rosenfeld Jun 20 '16 at 17:26
66
$\begingroup$

None of the answers mention that professional mathematicians don't specially go out of the way to convert everything to symbols. "$A$ is non-empty" is indeed the most common way to express the statement. Furthermore, for complicated structures it is almost always expressed this way, such as:

Given any non-empty chain of fields ordered by inclusion, their union is also a field.

$\endgroup$
  • 6
    $\begingroup$ My first thought as well, I feel this was seriously overlooked $\endgroup$ – Alex Mathers Jun 20 '16 at 13:31
  • 2
    $\begingroup$ @AlexMathers: Indeed, though it certainly wasn't overlooked by Hagen (see his comment). I'm of the opinion that in fact this is the right answer to What is usual way for professional mathematicians?. $\endgroup$ – user21820 Jun 20 '16 at 13:32
  • 2
    $\begingroup$ Is "non-empty" really more usual than "nonempty"? $\endgroup$ – bof Jun 21 '16 at 22:52
  • 2
    $\begingroup$ @user21820 You're entitled to your preferences. I prefer the unhyphenated forms. Save ink, paper, trees, forests, planets. $\endgroup$ – bof Jun 22 '16 at 7:10
  • 2
    $\begingroup$ @user21820 I have no idea what you mean by deeming a mathematical to be "grammatically invalid". In any case, this is mathematics, not poetry, so one should optimize the former, not the latter. $\endgroup$ – Bill Dubuque Jun 22 '16 at 13:10
10
$\begingroup$

$A \neq \varnothing$

[LaTeX: A \neq \varnothing]

$\endgroup$
  • $\begingroup$ Hmm, interesting. What's preferred to denote empty set: \emptyset or \varnothing? I'd suppose the former. $\endgroup$ – Ruslan Jun 20 '16 at 12:26
  • 7
    $\begingroup$ @Ruslan: \varnothing looks much nicer. \emptyset looks like a financial zero. $\endgroup$ – user21820 Jun 20 '16 at 13:22
  • 8
    $\begingroup$ @user21820: arguably, if you prefer the look of \varnothing but use it to denote empty sets, the semantically correct thing would be to \renewcommand{\emptyset}{\varnothing}. FWIW, I don't see why we need any symbol for the empty set at all. $\{\}$ is the perfect way to write it. $\endgroup$ – leftaroundabout Jun 20 '16 at 19:03
  • 1
    $\begingroup$ @leftaroundabout Perhaps for the same reason that $\vec{0}$ is a thing. The empty set is special enough to deserve its own zero-like symbol, because it is a zero. $\endgroup$ – Rhymoid Jun 22 '16 at 11:52
  • 1
    $\begingroup$ @leftaroundabout It's only as awkward as the context makes it. The ZF definition of natural numbers is definitely more awkward with $\varnothing$ than with $\left\{\right\}$. I can't come up with a good example, but I imagine that a context could demand that the zero vector is written out as $\left<0,0,0,\ldots\right>$. On the other hand, for the zero element of the monoid formed by $\mathcal{P}(X)$ and $\cup$, the better choice is $\varnothing$ or $\emptyset$ IMO, not $\left\{\right\}$. Just as in programming, intent matters when describing mathematical objects. $\endgroup$ – Rhymoid Jun 22 '16 at 19:47
6
$\begingroup$

$A \neq \emptyset$

Thats how it is commonly written

$\endgroup$
5
$\begingroup$

How about: $(\exists x)\, x\in A$? Alternatively, one could say that $A$ is inhabited. This usage avoids needless negation which is problematic constructively speaking, and is common in constructive mathematics.

$\endgroup$
  • 1
    $\begingroup$ To say to the reader that a function $f$ is continuous would you say $(\forall x \in \mathbb{R}) (\forall \varepsilon > 0) (\exists \delta > 0) \mid |t - x| < \delta \implies |f(t) - f(x)| < \varepsilon$? $\endgroup$ – MCT Jun 20 '16 at 15:15
  • 2
    $\begingroup$ Of course that's kind of a purposely bad example, but we define notions initially so we don't just have to say the definition every time we talk about it. This isn't so bad because it's short, but I don't think it's better than simply $A \neq \varnothing$ $\endgroup$ – MCT Jun 20 '16 at 15:15
  • $\begingroup$ The formula $\exists x\; x\in A$ has five symbols. That's not so bad compared to $|A|>0$ which has four. $\endgroup$ – Mikhail Katz Jun 20 '16 at 15:20
  • 12
    $\begingroup$ I don't see how using obscure terms from constructive logic is better when you just mean it to say "The set is not empty" (unless, of course, you're using it in a constructive logical context, which the question is not about). Yes, it's valid, but this question is about "the usual way for professional mathematicians." $\endgroup$ – MCT Jun 20 '16 at 15:28
  • 1
    $\begingroup$ I like the avoid needless negation point, but OTOH introducing an element $x$ without need is also not nice – unless you need such an element anyway! If you start with “Let $A\neq\{\}$ be a set and $x\in A$...”, then you could as well just make it “Let $A$ be a set with $x\in A$”, and it's clear that $A$ is nonempty. $\endgroup$ – leftaroundabout Jun 20 '16 at 19:11
3
$\begingroup$

Let $A$ denote a set.

In constructive mathematics, there's a difference between the statements '$A$ is non-empty,' which is defined to mean that $A$ is not isomorphic to $\emptyset$, and '$A$ is inhabited,' which is defined to mean that $A$ has at least one element, i.e. $\exists a \in A(\mathrm{True})$. Thus, depending on their standpoint and interests, a professional mathematician will probably write one of '$A$ is non-empty' or '$A$ is inhabited.'

Classically, these are equivalent.

$\endgroup$
  • $\begingroup$ Does constructive mathematics replace set-theoretic equality with isomorphism? What sort of isomorphism is it, then? My (poor) understanding of constructivist logic would have me believe that equality is not messed with. $\endgroup$ – 6005 Jun 23 '16 at 5:06
  • $\begingroup$ @6005, an isomorphism of sets is just a bijective function, and you're right that constructive mathematics doesn't mess with equality. But I prefer to avoid the material view in favour of the structural whenever I can. $\endgroup$ – goblin Jun 23 '16 at 10:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.