Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript describing the set.

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

defined as

The Iverson bracket allows the equivalent notation, ${\displaystyle [x\in A]}$, to be used instead of ${\displaystyle \mathbf {1} _{A}(x)}$.

The function ${\displaystyle \mathbf {1} _{A}}$ is sometimes denoted ${\displaystyle I_{A}}$, ${\displaystyle \chi _{A}}$, K_A or even just ${\displaystyle A}$. (The Greek letter ${\displaystyle \chi }$ appears because it is the initial letter of the Greek word χαρακτήρ, which is the ultimate origin of the word characteristic.)

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