# Does the function which unpacks singletons have a name?

Let $$X$$ be any set and $$P_1(X) = \{ \{x\} : x\in X\}$$. Does the function which unpacks the singleton, i.e.

$$f\colon P_1(X) \to X, \{x\} \mapsto x$$

have any special name?

On a related note, assume the set $$X$$ has some parition $$X=\bigcup_{i\in I} C_i$$ (where all $$C_i$$ are pairwise disjoint) How does one properly define the function $$p$$ which maps elements of $$X$$ onto the index $$i$$ of the partition they belong to? It seems to me the only way is to define

$$g: X\to P_1(I),\, x\mapsto \{i: x\in C_i\}$$

and then set $$p = f\circ g$$.

EDIT: I guess another way to write it is to use a conditional without an else clause

$$p\colon X\to I,\, x \mapsto \begin{cases}i: x\in C_i\end{cases}$$

which feels weird to say the least.

## 1 Answer

Defining $$g$$ and showing its range is a subset of $$P_1(I)$$ requires the same things as defining $$p$$ right away: namely that for every $$x$$ there's exactly one $$i\in I$$ such that $$x\in C_i$$.

Nevertheless, the function $$f$$ could be called the 'inverse of the canonical embedding $$X\to P(X)$$'.

• When I write 'proper' definition here, and I should have been more clear about that, I mean a symbolical, non-prose definition. Obviously one could just use define $p$ as the "the function that maps elements of $X$ onto the index $i$ of the partition they belong to" as I did in my OP. – Hyperplane Apr 17 at 14:18
• Well, the formal definition of a function $A\to B$ is that it's a set of ordered pairs $(a,b)$ such that $\forall a\exists! b: (a,b)$ is in the set. So, it can be made formal. – Berci Apr 17 at 14:22