What is the equivalent of "eval" function in math notations?

Question

Given two sets $X$ and $Y$, and a function $f : X \to Y$, would it be correct to write: \begin{equation} \forall y \in \{a,b,c\} : \textit{Fy} \triangleq \{ x \in X \mid f(x) = y \}. \end{equation} as an equivalent of: \begin{equation} \textit{Fa} \triangleq \{ x \in X \mid f(x) = a \}.\\ \textit{Fb} \triangleq \{ x \in X \mid f(x) = b \}.\\ \textit{Fc} \triangleq \{ x \in X \mid f(x) = c \}.\\ \end{equation}

I'm basically looking for an equivalent of the evaluation function.

N.B. I don't want to write $F_y$ instead of $Fy$.

Thanks for you answers.

NS

Answer 1

On usually writes

$$f^{-1}[y] := \{x\in X\mid f(x)=y\}.$$

It is called the "preimage" (or "inverse image"). You can compute the preimage of a set as well, i.e.

$$f^{-1}[\tilde Y]:=\{x\in X\mid f(x)\in \tilde Y\}.$$

Then $f^{-1}[y]$ is equivalent to $f^{-1}[\{y\}]$. When it is known that the resulting set contains exactly a single element (when $f$ is bijective/one-to-one), then one often writes $f^{-1}(y)$ instead, and associates the result with this single element. This is called the "inverse function".

Note: Taste might differ and the use of square brackets and parantheses for the preimage is not standard. I prefer the notation presented here, because it is clear whether the result is a set or a single element.