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

miracle173
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Light Yagmi
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    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) – JMoravitz Jun 20 '16 at 05:54
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    Actually, there is usually nothing wrong with writing "$A$ is not empty" in plain English (or whatever other language) in your text – Hagen von Eitzen Jun 20 '16 at 10:13
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    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. – Joonas Ilmavirta Jun 20 '16 at 15:07
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    I would go with the simplest way to express it and the most commonly known notation, just $A$ $\neq \emptyset $. – Iff Jun 20 '16 at 16:59
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    Did you just forget about the $\neq$ symbol since last year? – user2357112 Jun 20 '16 at 17:59
  • Perhaps we need a symbol crossing out the emptyset such as $\backslash !!! \emptyset$, similar to the UK "End of No Parking Zone" sign – Henry Jun 21 '16 at 15:28
  • also "$A$ is non-vacuous" – Andrea Jun 21 '16 at 19:02
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    When you write $A ! = \emptyset$, I may perversely read it as: $A$ factorial is the empty set. – GEdgar Jun 21 '16 at 22:40
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    When you edit a post (a question or an answer), just fix* it*.  Don’t fix it but then add a note restating the error and drawing attention to the fact that you fixed it.  Think of it as a paper that you’ve submitted to a journal or a conference — you wouldn’t include a section detailing problems with previous versions of the paper.  We can always look at the revision history of a post. – Scott - Слава Україні Jun 22 '16 at 08:36

6 Answers6

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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.

Eric Wofsey
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    +1 for several reasons, one explicitly pointing out that mathematics has it's own version of "!=" and not just implicitly using it. – pjs36 Jun 20 '16 at 06:14
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    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. – Ruslan Jun 20 '16 at 07:56
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    @Ruslan Some languages use a different ASCII approximation that is /= or =/= which is even more similar to the mathematical version. – Bakuriu Jun 20 '16 at 11:12
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    Also, \ne is the same as \neq. – user21820 Jun 20 '16 at 13:21
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    @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). – user21820 Jun 20 '16 at 14:07
  • Does $|A| > 0$ make sense if $A$ can be infinite? I personally don't know about how the $>$ operation extends to cardinals $\aleph_0$ and things like that – MT_ Jun 20 '16 at 15:12
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    Also, using $\varnothing$ (\varnothing) instead of $\emptyset$ (\emptyset) is a possible discussion. – JnxF Jun 20 '16 at 15:31
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    @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? – timur Jun 20 '16 at 15:37
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    @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. – Meni Rosenfeld Jun 20 '16 at 17:26
  • @Soke . Set-theorists use |A| to denote the cardinal of the set A, which, when A is finite, is the number of members of A.The topic of relations between cardinals of infinite sets can be found in any introductory book on set theory, and there are many many such books. – DanielWainfleet Jun 20 '16 at 22:41
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    One reason not to use $|A|> 0$ is that the absolute-value symbol has other uses, depending on context, and in a paper not directed to mathematicians, this might confuse the readers. – DanielWainfleet Jun 20 '16 at 22:44
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    @timur: Fortran is the one that used /= for "not-equal" (https://en.wikipedia.org/wiki/Fortran_95_language_features#Scalar_relational_operations). So there is no reason why the use of ! corresponds to the strike in "$\ne$". C's ! and != both come from B, but I couldn't trace the history beyond that. However, semantically it makes no sense to go backwards from != to ! since both seem to appear at the same time in programming language history. – user21820 Jun 21 '16 at 03:03
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    @user21820: I'd reckon it likely enough that ! and != were in fact chosen as a consistent pair right from the beginning. The following reasoning seems quite plausible to me: 1. Fortran's .NOT. is awkward, we should give negation a dedicated symbol! 2. there's already a “negation symbol” in /=. 3. Huh, but we can't very well use / as negation since that's already division! 4. Let's use ! then, as it looks somewhat similar. 5. For consistency, use it in both standalone negation, and negated equation, so this is now !=. – leftaroundabout Jun 22 '16 at 19:36
  • @leftaroundabout: I too believe that they were chosen simultaneously, but as I said I couldn't trace where either of them came from before B. I somehow doubt that similarity to a strike had much to do with it. What if it was simply close to the ~ key (which was used for bitwise negation)? (See http://naildrivin5.com/blog/2013/08/29/a-real-keyboard-for-programmers.html about the Datapoint 3300 which was around when B was created.) – user21820 Jun 23 '16 at 03:06
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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.

user21820
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    My first thought as well, I feel this was seriously overlooked – Alex Mathers Jun 20 '16 at 13:31
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    @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?. – user21820 Jun 20 '16 at 13:32
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    But often the empty-set is a degenerate case that plays no essential role e.g. the Lemma here, so having concise notation for this exclusion helps one to avoid obfuscating the essence of the matter with trivial details. – Bill Dubuque Jun 21 '16 at 22:43
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    Is "non-empty" really more usual than "nonempty"? – bof Jun 21 '16 at 22:52
  • @BillDubuque: Your example is grammatically invalid and hence it in fact shows that it is better to use "non-empty" instead of symbols. – user21820 Jun 22 '16 at 06:02
  • @bof: For this special case of "non-X", it is true that "nonempty" is more common, but I prefer the hyphenated version, such as in "non-negative" and "non-invertible" and "non-increasing". – user21820 Jun 22 '16 at 06:10
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    @user21820 You're entitled to your preferences. I prefer the unhyphenated forms. Save ink, paper, trees, forests, planets. – bof Jun 22 '16 at 07:10
  • @bof: Haha sure, by all means use whichever you prefer. I just like to be consistent and it seems that the prevalence of the hyphen seems to vary between different words. And I don't print, so I save everything you want. =) – user21820 Jun 22 '16 at 07:19
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    @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. – Bill Dubuque Jun 22 '16 at 13:10
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    @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. – user21820 Jun 22 '16 at 14:03
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    @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. – user21820 Jun 22 '16 at 14:27
  • @user21820 fyi: One of the primary goals of many of my posts is to teach students how to think like an algebraist or number-theorist (not how to write natural language like a poet). Let's consider an analogy. When you first learn a foreign language, you often have to translate back-and-forth between that and your native language. But once you become proficient, such crutches are no longer necessary. The same holds true for mathematical language ... – Bill Dubuque Jun 22 '16 at 14:57
  • ... Mathematics is independent of natural language (e.g. I have programmed computer algebra systems to do sophisticated mathematics but they do not understand any natural language). And some of my posts here contain little if any natural language.Hurray for the language of mathematics. For without it, the essence of most mathematical exposition would be extremely obfuscated. – Bill Dubuque Jun 22 '16 at 14:57
  • @BillDubuque: I understand what you're getting at, but there is a limit to how much people (not just students) can think in purely symbolic form. The very example you linked to is 90% English, not symbols, so your argument fails completely. If you look at the complete proofs in my posts you'll find that they are much more formal and consistent than yours, so I can say that I'm more committed to a formal proof style than you. If you don't mind my feedback, the problem with your style is the ad-hoc inconsistency in your notation and terminology, which simply cause readers difficulty. – user21820 Jun 22 '16 at 15:24
  • @user21820 There is no "failure". Rather, there is a different choice of what is deserving of emphasis, etc. And the point I made has little to do with "formality". In any case we are now very off-topic, and one could write volumes on such matters. so let's agree to disagree. – Bill Dubuque Jun 22 '16 at 15:34
  • @BillDubuque: Okay sure! – user21820 Jun 22 '16 at 16:27
  • PS Re: 90% English, I just posted a proof whose only English is "So". Once one is proficient reading such concise proofs, one can grok their essence in single glance, which would be impossible if it was obfuscated by gratuitous verbosity. In any case, I do understand and respect your viewpoint. Thankfully this forum welcomes all kinds of proofs, where they be concise or verbose. Enjoy. – Bill Dubuque Jun 22 '16 at 22:02
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$A \neq \varnothing$

[LaTeX: A \neq \varnothing]

  • Hmm, interesting. What's preferred to denote empty set: \emptyset or \varnothing? I'd suppose the former. – Ruslan Jun 20 '16 at 12:26
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    @Ruslan: \varnothing looks much nicer. \emptyset looks like a financial zero. – user21820 Jun 20 '16 at 13:22
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    @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. – leftaroundabout Jun 20 '16 at 19:03
  • @leftaroundabout: I actually create a new command \none because I don't like typing so much and don't like memorizing how much I've to type before the auto-complete gets the right command. – user21820 Jun 21 '16 at 02:24
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    @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. – Rhymoid Jun 22 '16 at 11:52
  • @Rhymoid how would you write the zero vector if not with a zero symbol? It would be very awkward, much unlike ${}$. – leftaroundabout Jun 22 '16 at 19:29
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    @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. – Rhymoid Jun 22 '16 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.

Mikhail Katz
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    To say to the reader that a function $f$ is continuous would you say $(\forall x \in \mathbb{R}) (\forall \varepsilon > 0) (\exists \delta > 0) \mid |t - x| < \delta \implies |f(t) - f(x)| < \varepsilon$? – MT_ Jun 20 '16 at 15:15
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    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 $A \neq \varnothing$ – MT_ Jun 20 '16 at 15:15
  • The formula $\exists x; x\in A$ has five symbols. That's not so bad compared to $|A|>0$ which has four. – Mikhail Katz Jun 20 '16 at 15:20
  • I don't think $|A| > 0$ is good either, which is why I didn't mention it. Imagine having to actually type either of those in an actual paper. How would you say it? Let $A$ be a set such that $\exists x \mid x \in A$ $\dots$? – MT_ Jun 20 '16 at 15:22
  • I would say "inhabited". – Mikhail Katz Jun 20 '16 at 15:23
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    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." – MT_ Jun 20 '16 at 15:28
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    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. – leftaroundabout Jun 20 '16 at 19:11
  • This is a very poor way to say that $A$ is nonempty. The word choice 'inhabited' doesn't sound good either. – Cyriac Antony Jan 05 '19 at 09:14
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$A \neq \emptyset$

Thats how it is commonly written

Zelos Malum
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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.

goblin GONE
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  • 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. – Caleb Stanford Jun 23 '16 at 05:06
  • @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. – goblin GONE Jun 23 '16 at 10:21