How do I reference the only element in a singleton set?

Question

You can turn any element $n$ into a singleton set by adding braces, $\{n\}$. Is there an inverse to this operation, such that if I know the set is a singleton set I can easily reference its element?

For example, say I have the set $A = \{5,6,7,8\}$, and then some process that iteratively eliminates all but one element of $A$, and then I want to see what 10 plus the resulting element is, how could I write that?

The only way I can think of is by referencing the first element by its index, e.g., $10 + A_1$. Is there another way to do it though? In my case the elements of $A$ are already indices, so I would have to index an index, like $10 + x_{A_1}$. Which is not the worst thing in the world, I was just wondering if there is a better way.

Answer 1

You could simply define a function

$\text{the} : \wp(X) \to X : \{x\} \mapsto x$

that maps every singleton set to its only element (the element $x$, which is the only member of $\{x\}$).

Then e.g. $\text{the}(\{5\})+10 = 5+10 = 15$.

I have not seen this notation used anywhere in practice, though.

Answer 2

A singleton set is a set with only one element. Thus, we can say "the element of" that set without error, because every element of $\{ a \}$ is equal to $a$.

We can use the Description operator :

"Following the example of Principia Mathematica, it is customary to use a definite description operator symbolized using the "turned" (rotated) Greek lower case iota character "$℩$".

The notation $℩ x(\phi x)$ means "the unique $x$ such that $\phi x$".

Thus :

$a= ℩ x( x \in \{ a \})$.