In the mathematical subfield of set theory, the **indicator function** is a function defined on a set *X* which is used to indicate membership of an element in a subset *A* of *X*.

**Remark**. The indicator function is sometimes also called *characteristic function*, although this usage is much less frequent now. The term "*characteristic function*" is also used in probability theory where it has an entirely different meaning, see characteristic function.

The indicator function of a subset *A* of a set *X* is a function

defined as

The indicator function of *A* is sometimes denoted

## Basic properties

The mapping which associates a subset *A* of *X* to its indicator function 1_{A} is injective; its range is the set of functions *f*:*X* →{0,1}.

If *A* and *B* are two subsets of *X*, then

More generally, suppose *A*_{1}, ..., *A*_{n} is a collection of subsets of *X*. For any *x* ∈ *X*,

is clearly a product of 0s and 1s. This product has the value 1 at precisely those *x* ∈ *X* which belong to none of the sets *A*_{k} and is 0 otherwise. That is

Expanding the product on the left hand side,

where |*F*| is the cardinality of *F*. This is one form of the principle of inclusion-exclusion.

As suggested by the previous example, the indicator function is a useful notational device in combinatorics. The notation is used in other places as well, for instance in probability theory: if *X* is a probability space with probability measure *P* and *A* is a measurable set, then *1*_{A} becomes a random variable whose expected value is equal to the probability of *A*:

Last updated: 05-07-2005 06:36:12

Last updated: 05-13-2005 07:56:04