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.

Definition

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

defined as

\begin{cases}
1 &\mbox{if}\ x \in A, \\
0 &\mbox{if}\ x \notin A.
\end{cases}

The Iverson bracket allows the notation $[x\; \backslash in\; A]$.

The indicator function of $A$ is sometimes denoted

(The Greek letter χ because it is the initial letter of the Greek etymon of the word characteristic.)

Remark on notation and terminology

• The notation $\backslash mathbf\{1\}\_A$ may signify the identity function.

• The notation $\backslash chi\_\{A\}$ may signify the characteristic function in convex analysis.

A related concept in statistics is that of a dummy variable (this must not be confused with "dummy variables" as that term is usually used in mathematics, also called a bound variable).

The term "characteristic function" has an unrelated meaning in probability theory. For this reason, probabilists use the term indicator function for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term characteristic function to describe the function which indicates membership in a set.

Basic properties

The mapping which associates a subset $A$ of $X$ to its indicator function $\backslash mathbf\{1\}\_A$ is injective; its range is the set of functions $f\; :\; X\; \backslash to\; \backslash \{0,1\backslash \}$.

In the following, the dot represents multiplication, 1·1 = 1, 1·0 = 0 etc. "+" and "−" represent addition and subtraction. "$\backslash cap$" and "$\backslash cup$" is intersection and union, respectively.

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

and the "complement" of the indicator function of A i.e. A^C is:

More generally, suppose $A\_1,\; \backslash ldots,\; A\_n$ is a collection of subsets of $X$. For any
$x\; \backslash in\; X$,

is clearly a product of $0$s and $1$s. This product has the value 1 at
precisely those $x\; \backslash in\; X$ 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 $\backslash mathbb\{P\}$ and $A$ is a measurable set, then $\backslash mathbf\{1\}\_A$ becomes a random variable whose expected value is equal to the probability of $A:$
This identity is used in a simple proof of Markov's inequality.

In many cases, such as order theory, the inverse of the indicator function may be defined. This is commonly called the generalized Möbius function, as a generalization of the inverse of the indicator function in elementary number theory, the Möbius function. (See paragraph below about the use of the inverse in classical recursion theory.)

Mean, Variance and covariance

Given a probability space: $\backslash textstyle\; (\backslash Omega,\; \backslash mathcal\; F,\; P)$ with A $\backslash in\; F$, the indicator random variable $\backslash mathbf\{1\}\_A\; :\; \backslash Omega\; \backslash rightarrow\; \backslash Bbb\{R\}$, is defined by $\backslash mathbf\{1\}\_A\; (\backslash omega)\; =\; 1$ if $\backslash omega\; \backslash in\; A$ otherwise $\backslash mathbf\{1\}\_A\; (\backslash omega)\; =\; 0$ (i.e basically $\backslash mathbf\{1\}\_A$ is a indicator random variable)

Characteristic function in recursion theory, Gödel's and Kleene's representing function

Kurt Gödel described the representing function in his 1934 paper "On Undecidable Propositions of Formal Mathematical Systems". (The paper appears on pp. 41-74 in Martin Davis ed. The Undecidable):
Stephen Kleene (1952) (p. 227) offers up the same definition in the context of the primitive recursive functions as a function φ of a predicate P, takes on values 0 if the predicate is true and 1 if the predicate is false.

For example, because the product of characteristic functions φ₁*φ₂* . . . *φ_n = 0 whenever any one of the functions equals 0, it plays the role of logical OR: IF φ₁=0 OR φ₂=0 OR . . . OR φ_n=0 THEN their product is 0. What appears to the modern reader as the representing function's logical-inversion, i.e. the representing function is 0 when the function R is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY (p. 228), the bounded- (p. 228) and unbounded- (p. 279ff) mu operators (Kleene (1952)) and the CASE function (p. 229).

Characteristic function in fuzzy set theory

In classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). In fuzzy set theory, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in some algebra or structure (usually required to be at least a poset or lattice). Such generalized characteristic functions are more usually called membership functions, and the corresponding "sets" are called fuzzy sets. Fuzzy sets model the gradual change in the membership degree seen in many real-world predicates like "tall", "warm", etc.

See also

• Free variables and bound variables

• Simple function

• Multiset

• Membership function

• Extension (predicate logic)