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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: $|A|\ != \emptyset \rightarrow A\ != \emptyset$

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    $\begingroup$ 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
    Commented Jun 20, 2016 at 5:54
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    $\begingroup$ 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
    Commented Jun 20, 2016 at 10:13
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    $\begingroup$ 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
    Commented Jun 20, 2016 at 15:07
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    $\begingroup$ I would go with the simplest way to express it and the most commonly known notation, just $A$ $\neq \emptyset $. $\endgroup$
    – Iff
    Commented Jun 20, 2016 at 16:59
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    $\begingroup$ Did you just forget about the $\neq$ symbol since last year? $\endgroup$
    – user2357112
    Commented Jun 20, 2016 at 17:59

6 Answers 6

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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.

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    $\begingroup$ +1 for several reasons, one explicitly pointing out that mathematics has it's own version of "!=" and not just implicitly using it. $\endgroup$
    – pjs36
    Commented Jun 20, 2016 at 6:14
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    $\begingroup$ 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
    Commented Jun 20, 2016 at 7:56
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    $\begingroup$ @Ruslan Some languages use a different ASCII approximation that is /= or =/= which is even more similar to the mathematical version. $\endgroup$
    – Bakuriu
    Commented Jun 20, 2016 at 11:12
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    $\begingroup$ @Ruslan: By the way != is not an ASCII approximation of "$\ne$" at all. It's because ! was used for logical negation in C. x!=y is equivalent to !(x==y). $\endgroup$
    – user21820
    Commented Jun 20, 2016 at 14:07
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    $\begingroup$ @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
    Commented Jun 20, 2016 at 15:37
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None of the other answers so far 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.

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    $\begingroup$ My first thought as well, I feel this was seriously overlooked $\endgroup$
    – Alex Mathers
    Commented Jun 20, 2016 at 13:31
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    $\begingroup$ @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
    Commented Jun 20, 2016 at 13:32
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    $\begingroup$ @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
    Commented Jun 22, 2016 at 13:10
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    $\begingroup$ @BillDubuque: Your last comment is not grammatically valid either, and poetry has nothing to do with grammar. My point is that what you wrote "Let $S \ne \varnothing$ be a set of integers ..." which does not correspond to a smooth grammatical English sentence. It is far better to write "Let $S$ be a non-empty set of integers ...". After all, using the symbols just makes it harder to process, not easier, contrary to your claim that concise notation for the empty-set is not obfuscating. If you disagree, then that's your opinion, not mine. $\endgroup$
    – user21820
    Commented Jun 22, 2016 at 14:03
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    $\begingroup$ @BillDubuque: Well I merely pointed out your error, and never used it as an argument. Don't make false assumptions. Certainly people can write however they want, but if you want to convey your mathematics clearly then I don't think your style helps at all. I've seen your answers here on Math SE, and frankly they are among the most obfuscated of those of the experts here, in my personal opinion. If you improve your style I'm confident that many people will appreciate them much more. $\endgroup$
    – user21820
    Commented Jun 22, 2016 at 14:27
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$A \neq \varnothing$

[LaTeX: A \neq \varnothing]

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  • $\begingroup$ Hmm, interesting. What's preferred to denote empty set: \emptyset or \varnothing? I'd suppose the former. $\endgroup$
    – Ruslan
    Commented Jun 20, 2016 at 12:26
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    $\begingroup$ @Ruslan: \varnothing looks much nicer. \emptyset looks like a financial zero. $\endgroup$
    – user21820
    Commented Jun 20, 2016 at 13:22
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    $\begingroup$ @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
    Commented Jun 20, 2016 at 19:03
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    $\begingroup$ @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
    Commented Jun 22, 2016 at 11:52
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    $\begingroup$ @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
    Commented Jun 22, 2016 at 19:47
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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.

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    $\begingroup$ To say to the reader that a function $f$ is continuous would you say $(\forall x \in \mathbb{R}) (\forall \varepsilon > 0) (\exists \delta > 0) \mid |t - x| < \delta \implies |f(t) - f(x)| < \varepsilon$? $\endgroup$
    – MT_
    Commented Jun 20, 2016 at 15:15
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    $\begingroup$ 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 $A \neq \varnothing$ $\endgroup$
    – MT_
    Commented Jun 20, 2016 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
    Commented Jun 20, 2016 at 15:20
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    $\begingroup$ 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." $\endgroup$
    – MT_
    Commented Jun 20, 2016 at 15:28
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    $\begingroup$ 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
    Commented Jun 20, 2016 at 19:11
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$A \neq \emptyset$

Thats how it is commonly written

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Caleb Stanford
    Commented Jun 23, 2016 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 GONE
    Commented Jun 23, 2016 at 10:21

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