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Given a set $A$, there is a function called its "characteristic function", usually denoted $\chi_A$, $\mathbf{1}_A$, $I_A$, or $K_A$, defined as follows:

$$ \chi_A(x) = \begin{cases} 1 &\text{if } x \in A, \\ 0 &\text{if } x \notin A. \end{cases} $$

Conversely, given some function $a$ such that for all $x$, $a(x)\in\{0,1\}$, we can use it as a characteristic function to construct a set:

$$?_a=\{x\mid a(x)=1\}$$

Does the set induced by $a$ have a commonly used name or symbol? Or is there even just a more concise notation to construct it?

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  • $\begingroup$ The support, or the preimage of 1. But in this very special case you should probably draw attemtion to the assumption. $\endgroup$ – Ian Aug 9 '18 at 17:16
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    $\begingroup$ You should include an explicit statement of your terminology. Something along the lines of "We write $X_a$ to denote the subset of $X$ given by $\{x\in X\mid a(x)=1\}$." or something like that. $\endgroup$ – Asaf Karagila Aug 9 '18 at 18:39
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This is called the support of $f$. It comes up quite a it in analysis.

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    $\begingroup$ I would still add an explicit explanation of the terminology. This is not as standard in set theoretic contexts. $\endgroup$ – Asaf Karagila Aug 9 '18 at 18:38

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