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I was wondering what the notations are for indicating the element of a singleton (or unit set, or set with cardinality 1). This would be the inverse of set construction:

$$X = \{y\} \tag{1}$$ $$y = \text{? } X \text{ ?} \tag{2}$$

I haven't seen examples of it, but I think using a notation like $X_1$ or $X_0$ is misleading. The general case of $X$ may not even be countable, even though it is obviously countable when $|X|=1$. For example, if $M$ is a set of sets of real numbers:

$$\forall X \in M \,:\,|X|=1 \Rightarrow P(X_0) \tag{3}$$

This seems like a possibility but since all of the $X$ aren't countable it looks misleading.

I found this post that used a notation $$y = \iota X \tag{4}$$

Linguistically it seems similar to the English article "the". I would probably read the above as "y equals the X".

I don't know how commonly used or recognized that notation is. Are there any other notations, possibly more common?

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DanielV
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    I don't know of a standard convention, but if it's established that $\lvert X \rvert = 1$, then $y \in X$ is pretty unambiguous. – MartianInvader Jul 01 '14 at 00:01
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    I would be careful using notation like $\iota X$ for some kind of definite description. In older works, and in a small corner of modern works, $\iota X$ is just a different notation for ${X}$. And if you were using an upright iota for definite descriptions, $\iota X$ is an incomplete fragment of that notation. – Malice Vidrine Jul 01 '14 at 00:16
  • It might help to think of the singleton $X$ containing only the element $y$ as being the set such that $\forall a:[a\in X \iff a=y]$. Then, of course, $y\in X$ by the reflexive property of equality. – Dan Christensen Jul 01 '14 at 04:18
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    A proposal: Use inverted braces to "unpack" $y$, like so: $\ y=}X{\ $. – Christian Blatter Jul 01 '14 at 10:00
  • @ChristianBlatter Heh that would really make parsing a nightmare. Also I prefer not to introduce new notations, just to build upon what exists. Andreas is about what I would prefer, something simple and obvious like $\text{The}(X)$ – DanielV Jul 01 '14 at 10:07
  • The inverted braces are attractive for international use, avoiding the English roots of "The". However, I agree we've gone too far with our reliance on computers to willfully introduce a parsing problem of that magnitude. – durette Jun 07 '18 at 01:08

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If you want to be strict with set theoretic context, then $y=\bigcup\{y\}=\bigcup X$. But this might not work very well outside of set theoretic contexts.

In the case that $X$ is a subset of an ordered set, then $y=\min X=\max X$ as well. There's probably no good, and general notation for this. But I honestly don't see why we would need one.

Asaf Karagila
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As Asaf said, in contexts (like ZFC) where everything is a set, you can use $\bigcup X$. Unfortnately, I'd expect that only set theorists will recognize what you're doing without further explanation. I've used the notation $\text{TheUnique}(X)$, but that was in a paper close to computer science, where multi-letter symbols like that are fairly common.

Andreas Blass
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I wandered here off the street looking for this answer, and from my further searching, this is the best I've got. The uniqueness quantification is a fairly concise way to show it's a unitary set.

The left hand side might need to be wrapped in an existential quantification. Please edit as necessary; I'm only an engineer, not a mathematician.

$$ x = ValueOf(X) \Longleftrightarrow \exists! x : x \in X $$

durette
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    I think that as an engineer you should know not to name things "Value Of"...and there really is no advantage of using $\exists !~ x : x \in X$ over just $X = {x}$. Also notice that $x$ is bound in the quantifier on the right but onbound on the left, so they can't be referring to the same thing, so it isn't quite right. Thanks for the interest but I think this is not a good direction. – DanielV Jun 07 '18 at 01:31
  • My backup plan was closer to.... $$ \exists (x = ValueOf(X)) \Longleftrightarrow \exists! x : x \in X $$ but that notation is really dirty – durette Jun 07 '18 at 02:26
  • In the relational algebra used to describe databases, it's not uncommon to see $R={r}$ with the understanding that $r$ means every $r$. It's sometimes not very rigorous. The answer to the question above will help me write clearly about an idea... – durette Jun 07 '18 at 04:23
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It's probably massive overkill, but you can use Bourbaki's tau, which picks an arbitrary element within a set or an arbitrary element satisfying some predicate. If the set is empty or the predicate is never true, it returns something arbitrary. This operator is also sometimes written $\varepsilon$. You can slightly expand it to operate on sets directly rather than or in addition to well-formed formulas.

$$ \tau(X) \in X \;\;\; \text{if and only if $X$ is non-empty} $$ $$ \tau(X) \not\in X \;\;\; \text{if and only if $X$ is empty} $$

Using this thing requires the axiom of global choice, which may or may not be a problem.

Also, nothing stops you from making your own notation and then explicitly calling it out as nonstandard and defining it if there's no widely-used notation available.

$$\mathop{\text{el}}(X) = \text{the unique $x$ in $X$ is $X$ is a singleton}$$

$$\mathop{\text{el}}(X) \;\;\;\text{is undefined if $X$ is not a singleton} $$

Greg Nisbet
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