Suppose $Y=\{y_1,\ldots,y_m\}$ partitions the set $X=\{x_1,\ldots,x_n\}$. I would like to define a function $y: X \to Y$ which returns $y \in Y$ if and only if $x \in y$. Is there a way to write this more formally?
I came up with
$$y(x) = \{y \in Y :y \ni x\}$$
or
$$y(x) = \{y:y \in Y \ \land \ x \in y \}$$
but I am not sure this makes sense.
(This answer just collects other responses into what I hope is a more coherent, single answer, with a few of my opinions added in.)
@LeanderTilstedKristensen 's second comment, "Also $\{y\in Y| x \in y \}$ is not an element of Y, but rather a subset of Y", is important - it means that your symbolic notations are actually incorrect: they specify a subset of $Y$, not an element of $Y$.
I also think that words are fine, and "$y(x)$ is the (unique) element of $Y$ that contains $x$" seems really clear, and anyone who has seen a partition before will understand it and know that it is a valid definition.
Of course, if a notation already exists we should use that instead of inventing something new. And if your audience is comfortable with the fact that partitions and equivalence relations are more-or-less the same thing, then @MarekKryspin 's suggestion is the best: write $[x]$ instead of $y(x)$.
If you might be using more than one partition, or really want to emphasize that the equivalence class derives from $Y$, the notation $[x]_Y$ would do the job. It's usually used as "$[x]_\sim$, where "$\sim$" is an equivalence relation, but again, partitions and equivalence relations are pretty interchangeable, so using $Y$ as the subscript would be clear.
The idea $$y(x) = \{y \in Y :y \ni x\}$$ is fine for me (but*). Since $Y$ is a partitions elements of it are not $\emptyset$ and for all $x\in X$ there are only one $y\in Y$ such that $x\in y$. If you like equivalence relation there is one connected with partition. So if $\sim\subset X\times X$ would be a equivalence relation such that $X/_\sim=Y$ your function could be represented as mapping $$X \ni x\mapsto [x]_{\sim}\in X/_\sim. $$
Edit*: Of course $\{y \in Y :y \ni x\}$ is a singleton of our point of interest as it was pointed out. I missed that sorry and thanks to @LeanderTilstedKristensen. However if you really want be fancy you can write $$y(x) = \bigcup\{y \in Y :y \ni x\}$$ in the spirit of descriptive set theory. But this is more like joke than real application thing.
