Element of a Singleton (set with one element) notation

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

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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.

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

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