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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.

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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 $X\to P(X)$'.

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  • $\begingroup$ 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. $\endgroup$ – Hyperplane Apr 17 at 14:18
  • $\begingroup$ 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. $\endgroup$ – Berci Apr 17 at 14:22

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