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I would gladly use a notation for “$A$ is a finite subset of $B$”, like $$A\sqsubset B \text{ or } A\underset{fin}{\subset} B,$$ but I have never seen a notation for that. Are there any?

EDIT: While waiting for a future standard, I will use Joffan’s $\ddot{\subset}$ coded as 

$\newcommand{\finsub}[0]{\mathrel{\ddot{\subset}}}$
$A\finsub B$

$\newcommand{\finsub}[0]{\mathrel{\ddot{\subset}}}$ I will paste the new command in the first row, and then use \finsub, resulting in $A\finsub B$ which I will explain after first use in each text. I guess that is satisfying enough.

And really, as you define sets and functions in a text, you could as well define relations without standard notations.

Rócherz
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Lehs
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    $A = {a_1, \ldots, a_n} \subset B$ is one way. – Arthur Jan 27 '15 at 09:00
  • @Arthur: Yes this is sometimes the best, if you are interested in naming the elements. – Lehs Jan 27 '15 at 09:06
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    I like your second variant. I often write something like $A \underset{\smash{\scriptsize \text{finite}}}{⊂} B$ myself (which looks better when handwritten). I think this will be understood by everyone immediately, so I wouldn’t worry about using it. – k.stm Jan 27 '15 at 09:06
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    Maybe we should invent something. How about $A \ddot{\subset} B$ ? – Joffan Jan 27 '15 at 09:34
  • @Joffan: a good suggestion, easy to write and without line spacing! But perhaps not intuitive enough? – Lehs Jan 27 '15 at 11:54
  • @Joffan: How did you do it? :) OK: \ddot {\subset} – Lehs Jan 27 '15 at 12:16
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    $A\subset B,\ |A|\in\Bbb N$ – CiaPan Jan 27 '15 at 12:39
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    The notation $\subset\subset$ is very common for "relatively compact", which means "finite" in the discrete case. – Christian Blatter Jan 27 '15 at 13:08
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    The fact that despite the pervasiveness of finite subsets in mathematics, no standard notation has emerged, probably means that it's simply unneeded. I've seen $A \in \mathcal{P}_f(B)$ at some point, but it's far from being standard. IMO it's better to write something like that explicitly, instead of wanting to cram as much meaning as possible in one or two symbols... – Najib Idrissi Jan 27 '15 at 14:14
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    "Let $A$ be a finite subset of $B$" works perfectly well if you need this as part of a sentence. – Milo Brandt Jan 27 '15 at 18:09
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    Note: when LaTeXing, enclose your \ddot{\subset} in a \mathrel to get proper spacing as a relational operator: $A \ddot{\subset} B$ vs. $A \mathrel{\ddot{\subset}} B$. (Of course, you should \newcommand it, too.) – wchargin Jan 27 '15 at 20:28
  • $A\in\cup_{n\in\Bbb N} \binom{B}{n}$? But I'm more on the 'don't invent your own symbol for this' side. – Myself Jan 27 '15 at 22:44
  • @WChargin: thanks for the hint! – Lehs Jan 28 '15 at 07:52
  • I am searching for a symbol for finite sets too. There are other uses: finite unions, finite suprema. – beroal Apr 18 '17 at 11:01
  • Does this answer your question? Notation for the set of all finite subsets of $\mathbb{N}$ – user76284 Apr 19 '20 at 07:54

6 Answers6

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The usual way is to use two different notations, one of which means that $A$ is finite, and the other means it's a subset of $B$. $$A\subset B,\qquad|A|<\infty.$$

amWhy
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Amitai Yuval
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    Yes, but I don't like it. Especially since $\infty<\infty$. – Lehs Jan 27 '15 at 09:03
  • Moreover, $\mathbb N\subset\mathbb R$. – davcha Jan 27 '15 at 09:04
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    @Lehs What structure are you working in in which $\infty<\infty$ ? – Jack M Jan 27 '15 at 11:16
  • @JackM: like $\aleph_0<\aleph_1$? – Lehs Jan 27 '15 at 12:04
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    @Lehs: then the question becomes, why are you using a single symbol for two different sets? Naturally if $\infty$ stands for $\aleph_1$ then $|A|<\infty$ doesn't constrain $A$ to be finite, but that's because of bad notation to begin with :-) More conventionally, $<\infty$, if used at all, means "not infinite", i.e, "finite", but I certainly agree with comments and answers elsewhere that $<\omega$, $<\aleph_0$ or $\in\mathbb{N}$ are better. – Steve Jessop Jan 27 '15 at 12:56
  • @SteveJessop No. $<\omega$ is not better unless you're a set theorist. As you note yourself, $|A|<\infty$ is the most standard way of writing down that $A$ is finite, if you don't like "$A$ finite". That's what I would actually prefer myself: $A\subset B,\quad A \text{ finite}$. – yo' Jan 27 '15 at 15:40
  • I see that Steve Jessop already mentioned $ |A| \in \mathbb{N} $ which is close to what I would suggest to avoid confusion about infinities, but I'd use $ |A| \in \mathbb{N}_0 $ to make it clear that the empty set is allowed. There's no agreement over whether $ 0 \in \mathbb{N} $ – hvd Jan 28 '15 at 14:38
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    The symbol $\infty$ should never be used to denote a cardinal or an ordinal. It can, however, be used to denote something "not finite", and therefore $|A|<\infty$ is a perfectly valid way to denote finiteness. – Asaf Karagila Jan 28 '15 at 15:14
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There are two options:

  1. You use the notation often. Then define it properly at the beginning (or when you first need it) and use whatever you think is reasonable. I'd suggest, as others: $$ A \subset_{\mathrm{fin}} B, \quad A \sqsubset B, \quad A \mathrel{\ddot{\subset}} B, \quad A \subset\!\!\!\!\!\cdot\!\!\cdot\, B \quad \ldots $$

  2. You use it one or twice. Then just spell it out:

    • ... where $A\subset B$ is finite
    • ... $(\forall A \subset B, \, A\text{ finite})$
    • ...

Of course, you can use $|A|<\infty$ or $|A|<|\mathbb N|$ or $|A|<\omega$ or whatever, just try to imagine being a reader of your text and think what is the least confusing thing.

yo'
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In set theory there are two standard notations for the set of finite subsets of $X$:

  1. $[X]^{<\omega}$,
  2. $\mathcal P_\omega(X)$ or $\mathcal P_{\aleph_0}(X)$.

In naive set theory, you can also find $\operatorname{Fin}(X)$ quite often.

So if you'd want to write that $A$ is a finite subset of $B$, you could say that $A\in[B]^{<\omega}$ or $A\in\mathcal P_\omega(B)$ or $A\in\operatorname{Fin}(B)$.

Outside of set theory, I believe writing $|A|<\infty$ is probably one of the most accepted ways of writing that $A$ is finite. But I'm sure that in some fields of mathematics there are more or less common notations, and you should probably align yourself to the crowd which will read your work.

If you are writing for yourself, then it really doesn't matter what you're using, right?

In any case, do remember the standard advice about notation:

Notation is used to reduce clutter, not to abbreviate. Define your notations and be consistent with them. Don't force the reader to keep track of your symbols, if there's no need to do so.

Asaf Karagila
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  • It's is OK writing $A\subset B$ and then write ($A$ is finite) in the end of the row, but if there where some standard ($\subset\subset$?) I would prefer a notation that explained an important fact instantly. – Lehs Jan 27 '15 at 17:39
  • I don't know of any modifier to the $\subseteq$ symbol which denotes finiteness in any kind. – Asaf Karagila Jan 27 '15 at 17:46
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    @Lehs No notation is able to explain an important fact. Only words can. – yo' Jan 28 '15 at 00:03
  • @yo': Math is an extension of language and everything that can be expressed by symbols and formulas can also be expressed by words. So far so good. But the formula $A\subset B$ doesn't explain less than "$A$ is a subset of $B$" and it explain a fact good enough for me. The same would work for something lika $A\mathrel{\ddot{\subset}}B$. – Lehs Jan 28 '15 at 09:13
  • @Lehs: You seem to insist that you want this in terms of notations, instead of terms of clarity to the reader (unless the reader is you). Do whatever you want, just be sure to define the notation when you are setting the context, so people who are new to your conventions will be able to understand it. – Asaf Karagila Jan 28 '15 at 09:46
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    @Lehs Even though, if the fact that $A\subset B$ was particularly important and crucial for something, and not obvious at the moment in the current context, I would not hesitate much to say: "... because, from (3.14) we see that $A$ is a finite subset of $B$", or something in that manner. You are right that everything can be expressed in symbols, and it should be used as a good tool! Use symbol when appropriate, and accompany them with words when appropriate. – yo' Jan 28 '15 at 11:32
  • @yo': agree totally, but the "need" of $A\mathrel{\ddot{\subset}}B$ depend of the context. Certain formulas say more than words, and more instant. – Lehs Jan 28 '15 at 11:49
  • @Lehs: You need to take into consideration the fact that new notations are not immediately cached into the mind of the reader. So it is certainly doesn't have to be "more instant" than words. – Asaf Karagila Jan 28 '15 at 12:04
  • @AsafKaragila: I was pondering about the difference between a notation/formula visa vi using language. – Lehs Jan 28 '15 at 12:20
  • @Lehs: Formulas are clear when the notation is truly simplifying; the language is clear when there are not too many conjunctions, implications and negations. – Asaf Karagila Jan 28 '15 at 12:52
  • @AsafKaragila Where are you sourcing the powerset based notation? My vague memories of Jech made me think the subscript should be $< \omega$. Or is this motivated by some kind of forcing poset so you always want sets of size less than the given cardinal? – Peter Gerdes Mar 03 '21 at 23:46
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    @Peter: The very standard notation is $\mathcal P_\kappa(\lambda)$ for the subsets of $\lambda$ of size $<\kappa$. Here $\kappa=\omega$ or $\aleph_0$. – Asaf Karagila Mar 04 '21 at 00:32
  • Thanks, I phrased that badly. I didn't mean to suggest you were wrong merely express my surprise and ask for a ref to source it in my paper (we follow blah in...) and ask why the hell it is defined that way...anyway I managed to track it down to page 52 of jech. I mean, I guess it's the more flexible def since u can always add 1 but besides use as a forcing poset it still seems like a weird choice...then again your trees grow down so why not ;-) – Peter Gerdes Mar 04 '21 at 00:41
  • @PeterGerdes: It can't be a forcing poset, since it's a directed set. Directed sets are forcing equivalent to the trivial forcing. – Asaf Karagila Mar 04 '21 at 01:27
  • @AsafKaragila Yes under subset but that wasn't what I was thinking. I was thinking that it might be useful when specifying a poset eg Matthias conditions. I was just guessing at where having size less than would be so much more common than less than or equal you would build it into the notation. – Peter Gerdes Mar 04 '21 at 01:49
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If I had to invent a notation for this, the most suggestive I can think of is $$ A\in \mathcal P_{<\omega}(B). $$ One could define the RHS using another notation that in fact is more or less standard in some areas: $$ \mathcal P_{<\omega}(B) = \bigcup_{n \in \Bbb N}\binom Bn $$ which notation is derived from the that for the binomial coefficient similarly to the way the Cartesian product notation $A\times B$ is derived from that for multiplication, or $Y^X=\{\,f:X\to Y\,\}$ from exponentiation.

yo'
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Marc van Leeuwen
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Probably use ordinal/cardinal numbers?

$$|A| < \omega$$

$$|A| < \aleph_0$$

lisyarus
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$A\in[B]^{\lt\omega}$ where $[B]^{\lt\omega}$ denotes the set of all finite subsets of $B$.

bof
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  • This is better than $|A|<\infty$ but I would like to be able to write $A\ddot{\subset}B\subseteq \Bbb N$ ... – Lehs Jan 27 '15 at 12:40
  • In that case, I'd go with something like $A\underset{fin}{\subseteq}B\subseteq\mathbb N$. Easier on the reader than using some arbitrary symbol; easier to remember. – bof Jan 27 '15 at 12:57
  • It's excellent except for being to high and interacting on line spacing. – Lehs Jan 27 '15 at 13:00
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    Well then woule $A\subseteq_{fin}B$ work? – bof Jan 27 '15 at 13:16