# 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 χ_{A} or *I*_{A}, 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:

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

- In probability theory, the
**characteristic function**of any probability distribution on the real line is given by the following formula, where*X*is any random variable with the distribution in question:

- where "E" means expected value. See characteristic function (probability theory).

- The characteristic function in game theory.