# 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 specifying the subset.

In other contexts, such as computer science, this would more often be described as a **boolean predicate function** (to test set inclusion).

## Definition

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

defined as

The Iverson bracket allows the equivalent notation, , to be used instead of .

The function
is sometimes denoted
,
, *K _{A}* or even just
. (The Greek letter
appears because it is the initial letter of the Greek word χαρακτήρ, which is the ultimate origin of the word

*characteristic*.)

The set of all indicator functions on can be identified with , the power set of . Consequently, both sets are sometimes denoted by . This is a special case ( ) of the notation for the set of all functions .

## Remark on notation and terminology

- The notation
is also used to denote the identity function of
*A*. - The notation is also used to denote 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 classic probability theory. For this reason, traditional 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 that indicates membership in a set.

In fuzzy logic and modern many-valued logic, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.

## Basic properties

The *indicator* or *characteristic* function of a subset *A* of some set *X*, maps elements of *X* to the range {0,1}.

This mapping is surjective only when *A* is a non-empty proper subset of *X*. If *A* ≡ *X*, then
**1**_{A} = 1. By a similar argument, if *A* ≡ Ø then **1**_{A} = 0.

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

If and are two subsets of , then

and the indicator function of the complement of i.e. is:

- .

More generally, suppose
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* that 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 is a probability space with probability measure and is a measurable set, then becomes a random variable whose expected value is equal to the probability of :

- .

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 with , the indicator random variable is defined by if otherwise

## 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*):

- "There shall correspond to each class or relation R a representing function φ(x
_{1}, . . ., x_{n}) = 0 if R(x_{1}, . . ., x_{n}) and φ(x_{1}, . . ., x_{n}) = 1 if ~R(x_{1}, . . ., x_{n})." (p. 42; the "~" indicates logical inversion i.e. "NOT")

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 φ_{1}*φ_{2}* . . . *φ_{n} = 0 whenever any one of the functions equals 0, it plays the role of logical OR: IF φ_{1} = 0 OR φ_{2} = 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.

## Derivatives of the indicator function

A particular indicator function is the Heaviside step function. The Heaviside step function is the indicator function of the one-dimensional positive half-line, i.e. the domain [0, ∞). It is well known that the distributional derivative of the Heaviside step function, indicated by *H*(*x*), is equal to the Dirac delta function, i.e.

with the following property:

The derivative of the Heaviside step function can be seen as the 'inward normal derivative' at the 'boundary' of the domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domain *D*. The surface of *D* will be denoted by *S*. Proceeding, it can be derived that the inward normal derivative of the indicator gives rise to a 'surface delta function', which can be indicated by δ_{S}(**x**):

where *n* is the outward normal of the surface *S*. This 'surface delta function' has the following property:^{[1]}

By setting the function *f* equal to one, it follows that the inward normal derivative of the indicator integrates to the numerical value of the surface area *S*.

## See also

- Dirac measure
- Laplacian of the indicator
- Dirac delta
- Extension (predicate logic)
- Free variables and bound variables
- Heaviside step function
- Iverson bracket
- Kronecker delta, a function that can be viewed as an indicator for the identity relation
- Macaulay brackets
- Multiset
- Membership function
- Simple function
- Dummy variable (statistics)
- Statistical classification
- Zero-one loss function

## Notes

- ↑ Lange, Rutger-Jan (2012), "Potential theory, path integrals and the Laplacian of the indicator",
*Journal of High Energy Physics*, Springer,**2012**(11): 29–30, arXiv:1302.0864, Bibcode:2012JHEP...11..032L, doi:10.1007/JHEP11(2012)032

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*Introduction to Algorithms*(Second ed.). MIT Press and McGraw-Hill. pp. 94–99. ISBN 0-262-03293-7. - Davis, Martin, ed. (1965).
*The Undecidable*. New York: Raven Press Books, Ltd. - Kleene, Stephen (1971) [1952].
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