## Characteristic function

### Set

context | $X$ |

context | $A\subseteq X$ |

definiendum | $\chi_A:X\to \{0,1\} $ |

definiendum | $ \chi_A(x) := \begin{cases} 1 & \mathrm{if}\ x\in A\\\\ 0 & \mathrm{else} \end{cases}$ |

The complement $A^c$ is taken w.r.t. $X$.

### Discussion

The symbols “$0,1$” here are essentially placeholders. One could choose any set of two distinct elements as codomain.

#### Reference

Wikipedia: Characteristic function