How to write the set defined by an indicator function?

Question

For a set $A$ we can write its indicator function $\mathbf{1}_A$ as

$$\mathbf{1}_A(x) := \begin{cases} 1 & \text{if } x \in A\\ 0 & \text{if } x \notin A \end{cases}$$

Is there any agreed-upon notation for the reverse, the set characterized by the function $f$?

Answer 1

There is no particular notation, as far as I know. Usually you would just write $$f^{-1}(\{1\}).$$

Answer 2

(I'm not sure what you're referring to by $f$.)

If predicate $\mathbf 1_A(x)$ is defined in terms of set $A$, it implicitly has two parameters, set $A$ and element $x$. In that case, the predicate you're looking for could be expressed $\mathbf 1_{A^\prime}(x)$ (in other words, just pass in the complement of $A$ as a parameter to predicate $\mathbf 1$).

If membership of $A$ is determined by the predicate, in other words, if $A=\left\{x:\mathbf1_A(x)\right\}$, then predicate $\mathbf 1_A(x)$ is implicitly a boolean function, and the predicate determining membership to the complement of $A$ could just be written $\lnot \mathbf 1_A(x)$ or $\overline{\mathbf 1_A(x)}$, in other words, $A^\prime=\left\{x:\lnot \mathbf 1_A(x)\right\}$.