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In mathematics, an indicator function or a characteristic function of a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s.


Definition

The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just .


Notation and terminology

The notation \chi_A is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. 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 of some set maps elements of to the range \. This mapping is surjective only when is a non-empty
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of . If A \equiv X, then \mathbf_A=1. By a similar argument, if A\equiv\emptyset then \mathbf_A=0. In the following, the dot represents multiplication, 1\cdot1 = 1, 1\cdot0 = 0, etc. "+" and "−" represent addition and subtraction. "\cap " and "\cup " are intersection and union, respectively. If A and B are two subsets of X, then \begin \mathbf_ = \min\ = \mathbf_A \cdot\mathbf_B, \\ \mathbf_ = \max\ = \mathbf_A + \mathbf_B - \mathbf_A \cdot\mathbf_B, \end and the indicator function of the complement of A i.e. A^C is: \mathbf_ = 1-\mathbf_A. More generally, suppose A_1, \dotsc, A_n is a collection of subsets of . For any x \in X: \prod_ ( 1 - \mathbf_(x)) is clearly a product of s and s. This product has the value 1 at precisely those x \in X that belong to none of the sets A_k and is 0 otherwise. That is \prod_ ( 1 - \mathbf_) = \mathbf_ = 1 - \mathbf_. Expanding the product on the left hand side, \mathbf_= 1 - \sum_ (-1)^ \mathbf_ = \sum_ (-1)^ \mathbf_ where , F, is the cardinality of . 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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
. The notation is used in other places as well, for instance in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
: if is a probability space with probability measure \operatorname and is a measurable set, then \mathbf_A becomes a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
whose
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
is equal to the probability of : \operatorname(\mathbf_A)= \int_ \mathbf_A(x)\,d\operatorname = \int_ d\operatorname = \operatorname(A). This identity is used in a simple proof of
Markov's inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Marko ...
. In many cases, such as order theory, the inverse of the indicator function may be defined. This is commonly called the
generalized Möbius function In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural constructi ...
, as a generalization of the inverse of the indicator function in elementary
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, 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, \operatorname) with A \in \mathcal F, the indicator random variable \mathbf_A \colon \Omega \rightarrow \mathbb is defined by \mathbf_A (\omega) = 1 if \omega \in A, otherwise \mathbf_A (\omega) = 0. ;
Mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...
: \operatorname(\mathbf_A (\omega)) = \operatorname(A) (also called "Fundamental Bridge"). ;
Variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
: \operatorname(\mathbf_A (\omega)) = \operatorname(A)(1 - \operatorname(A)) ; Covariance: \operatorname(\mathbf_A (\omega), \mathbf_B (\omega)) = \operatorname(A \cap B) - \operatorname(A)\operatorname(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 "¬" indicates logical inversion, i.e. "NOT"): Kleene offers up the same definition in the context of the primitive recursive functions as a function of a predicate takes on values if the predicate is true and if the predicate is false. For example, because the product of characteristic functions \phi_1 * \phi_2 * \cdots * \phi_n = 0 whenever any one of the functions equals , it plays the role of logical OR: IF \phi_1 = 0 OR \phi_2 = 0 OR ... OR \phi_n = 0 THEN their product is . What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is when the function is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY, the bounded- and unbounded- mu operators and the CASE function.


Characteristic function in fuzzy set theory

In classical mathematics, characteristic functions of sets only take values (members) or (non-members). In '' fuzzy set theory'', characteristic functions are generalized to take value in the real unit interval , or more generally, in some
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
or structure (usually required to be at least a
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
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 Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
seen in many real-world predicates like "tall", "warm", etc.


Derivatives of the indicator function

A particular indicator function is the Heaviside step function H(x) := \mathbf_ The distributional derivative of the Heaviside step function is equal to the Dirac delta function, i.e. \frac=\delta(x) and similarly the distributional derivative of G(x) := \mathbf_ is \frac=-\delta(x) Thus 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 . The surface of will be denoted by . 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 \delta_S(\mathbf): \delta_S(\mathbf) = -\mathbf_x \cdot \nabla_x\mathbf_ where is the outward normal of the surface . This 'surface delta function' has the following property: -\int_f(\mathbf)\,\mathbf_x\cdot\nabla_x\mathbf_\;d^\mathbf = \oint_\,f(\mathbf)\;d^\mathbf. By setting the function equal to one, it follows that the inward normal derivative of the indicator integrates to the numerical value of the surface area .


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 In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
* Membership function * Simple function * Dummy variable (statistics) * Statistical classification * Zero-one loss function


Notes


References


Sources

* * * * * * * {{refend Measure theory Integral calculus Real analysis Mathematical logic Basic concepts in set theory Probability theory Types of functions