Let $A$ and $B$ be two disjoint sets (i.e. $A \cap B = \varnothing$), define $C = \{A, B\}$, and let $x$ be some value.
What is the operation that it will tell me which subset of $C$, if any, contains $x$?
Possible answers are $x \in A$, $x \in B$, or $x \notin \mathcal{P}(C)$. I suspect there is no such operation defined, but then how can this be written in formal mathematics?
There is no standard operation for this, but you could define an operation if you wanted to. For example, given a family $\mathcal{F}$ of pairwise disjoint sets and an object $x$, define $$\mathrm{find}_{\mathcal{F}}(x) = \bigcup \{ X \in \mathcal{F} \mid x \in X \}$$ Note that $\{ X \in \mathcal{F} \mid x \in X \}$ will necessarily be either empty or a singleton.
In the case where $\mathcal{F} = \{ A, B \}$ is a set containing just two disjoint sets $A$ and $B$, this will give $$\mathrm{find}_{\mathcal{F}}(x) = \begin{cases} A & \text{if } x \in A \\ B & \text{if } x \in B \\ \varnothing & \text{if } x \not\in A \text{ and } x \not\in B \end{cases}$$
Bit like I said, this is not standard notation, so if you wanted to use it then you'd need to define it first.