Indicator function

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The graph of the indicator function of a two-dimensional subset of a square.

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

${\mathbf {1}}_{A}\colon X\to \{0,1\}\,$

defined as

${\mathbf {1}}_{A}(x):={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x\notin A.\end{cases}}$

The Iverson bracket allows the equivalent notation, [x ∈ A], to be used instead of 1_A(x).

The function 1_A is sometimes denoted 1_{A ∈ A}, χ_A or I_A or even just A. (The Greek letter χ appears because it is the initial letter of the Greek word characteristic.)

Remark on notation and terminology

The notation 1_A is also used to denote the identity function.
The notation χ_A 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 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 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. " $\cap$ " and " $\cup$ " is intersection and union, respectively.

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

${\mathbf {1}}_{{A\cap B}}=\min\{{\mathbf {1}}_{A},{\mathbf {1}}_{B}\}={\mathbf {1}}_{A}\cdot {\mathbf {1}}_{B},$

${\mathbf {1}}_{{A\cup B}}=\max\{{{\mathbf {1}}_{A},{\mathbf {1}}_{B}}\}={\mathbf {1}}_{A}+{\mathbf {1}}_{B}-{\mathbf {1}}_{A}\cdot {\mathbf {1}}_{B},$

and the indicator function of the complement of $A$ i.e. $A^{C}$ is:

${\mathbf {1}}_{{A^{\complement }}}=1-{\mathbf {1}}_{A}$ .

More generally, suppose $A_{1},\dotsc ,A_{n}$ is a collection of subsets of X. For any x ∈ X:

$\prod _{{k\in I}}(1-{\mathbf {1}}_{{A_{k}}}(x))$

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

$\prod _{{k\in I}}(1-{\mathbf {1}}_{{A_{k}}})={\mathbf {1}}_{{X-\bigcup _{{k}}A_{k}}}=1-{\mathbf {1}}_{{\bigcup _{{k}}A_{k}}}.$

Expanding the product on the left hand side,

${\mathbf {1}}_{{\bigcup _{{k}}A_{k}}}=1-\sum _{{F\subseteq \{1,2,\dotsc ,n\}}}(-1)^{{|F|}}{\mathbf {1}}_{{\bigcap _{F}A_{k}}}=\sum _{{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}}(-1)^{{|F|+1}}{\mathbf {1}}_{{\bigcap _{F}A_{k}}}$

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 ${\mathbb {P}}$ and $A$ is a measurable set, then ${\mathbf {1}}_{A}$ becomes a random variable whose expected value is equal to the probability of $A$ :

$\operatorname {E}({\mathbf {1}}_{A})=\int _{{X}}{\mathbf {1}}_{A}(x)\,d{\mathbb {P}}=\int _{{A}}d{\mathbb {P}}=\operatorname {P}(A)$ .

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 $\textstyle (\Omega ,{\mathcal F},{\mathbb P})$ with $A\in {\mathcal F}$ , the indicator random variable ${\mathbf {1}}_{A}\colon \Omega \rightarrow {\mathbb {R}}$ is defined by ${\mathbf {1}}_{A}(\omega )=1$ if $\omega \in A,$ otherwise ${\mathbf {1}}_{A}(\omega )=0.$

Mean: $\operatorname {E}({\mathbf {1}}_{A}(\omega ))=\operatorname {P}(A)$

Variance: $\operatorname {Var}({\mathbf {1}}_{A}(\omega ))=\operatorname {P}(A)(1-\operatorname {P}(A))$

Covariance: $\operatorname {Cov}({\mathbf {1}}_{A}(\omega ),{\mathbf {1}}_{B}(\omega ))=\operatorname {P}(A\cap B)-\operatorname {P}(A)\operatorname {P}(B)$

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(x₁, . . ., x_n) = 0 if R(x₁, . . ., x_n) and φ(x₁, . . ., x_n) = 1 if ~R(x₁, . . ., 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 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.

Derivatives of the indicator function

A particular indicator function, which is very well known, 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.

$\delta (x)={\tfrac {dH(x)}{dx}},$

with the following property:

$\int _{{-\infty }}^{\infty }f(x)\,\delta (x)dx=f(0).$

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):

$\delta _{S}({\mathbf {x}})=-{\mathbf {n}}_{x}\cdot \nabla _{x}{\mathbf {1}}_{{{\mathbf {x}}\in D}}$

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

$-\int _{{{\mathbf {R}}^{n}}}f({\mathbf {x}})\,{\mathbf {n}}_{x}\cdot \nabla _{x}{\mathbf {1}}_{{{\mathbf {x}}\in D}}\;d^{{n}}{\mathbf {x}}=\oint _{{S}}\,f({\mathbf {\beta }})\;d^{{n-1}}{\mathbf {\beta }}.$

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.

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

References

Folland, G.B. (1999). Real Analysis: Modern Techniques and Their Applications (Second ed.). John Wiley & Sons, Inc.
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Section 5.2: Indicator random variables". Introduction to Algorithms (Second Edition 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]. Introduction to Metamathematics (Sixth Reprint with corrections). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company.
Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge UK: Cambridge University Press. ISBN 0-521-00758-5.
Zadeh, Lotfi A. (June 1965). "Fuzzy sets" (PDF). Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241-X.
Goguen, Joseph (1967). "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18 (1): 145–174. doi:10.1016/0022-247X(67)90189-8.

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