# Characteristic function  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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: $\chi _{\emptyset }=0;\,$ • Intersection: $\chi _{A\cap B}=\min\{\chi _{A},\chi _{B}\}=\chi _{A}\cdot \chi _{B};\,$ • Union: $\chi _{A\cup B}=\max\{\chi _{A},\chi _{B}\}=\chi _{A}+\chi _{B}-\chi _{A}\cdot \chi _{B};\,$ • complement: $\chi _{-A}=1-\chi _{A}$ • Inclusion: $A\subseteq B\Leftrightarrow \chi _{A}\leq \chi _{B}.\,$ In mathematics, characteristic function can refer also to any several distinct concepts:

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