# Characteristic function

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In set theory, the characteristic function or indicator function of a subset X of a set S is the function, often denoted χA or IA, from S to the set {0,1} which takes the value 1 on elements of X and 0 otherwise.

We can express elementary set-theoretic operations in terms of characteristic functions:

• Empty set: ${\displaystyle \chi _{\emptyset }=0;\,}$
• Intersection: ${\displaystyle \chi _{A\cap B}=\min\{\chi _{A},\chi _{B}\}=\chi _{A}\cdot \chi _{B};\,}$
• Union: ${\displaystyle \chi _{A\cup B}=\max\{\chi _{A},\chi _{B}\}=\chi _{A}+\chi _{B}-\chi _{A}\cdot \chi _{B};\,}$
• complement: ${\displaystyle \chi _{-A}=1-\chi _{A}}$
• Inclusion: ${\displaystyle A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}.\,}$

In mathematics, characteristic function can refer also to any several distinct concepts:

${\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}$
${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right)\,}$
where "E" means expected value. See characteristic function (probability theory).