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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an indicator function or a characteristic function of a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number). \mathbf 1_\Q(x) = \begin 1 ...
is the indicator function of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s as a subset of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
in
convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of s ...
, which is defined as if using the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of the standard definition of the indicator function. A related concept in
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
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 In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a Mathematical notation, notation (symbol) that specifies places in an expression (mathematics), expressi ...
.) The term "
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
" 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 Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
and modern many-valued logic, predicates are the characteristic functions of a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of a subset of some set
maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
elements of to the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
\. This mapping is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
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 of ...
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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 appl ...
. 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 In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
with probability measure \operatorname and is a
measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
, 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 po ...
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 l ...
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 (mathematics), function of a random variable is greater than or equal to some positive Constant (mathematics), constant. It is named a ...
. In many cases, such as
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, 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 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
. (See paragraph below about the use of the inverse in classical recursion theory.)


Mean, variance and covariance

Given a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
\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 ''arithme ...
: \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 numbers ...
: \operatorname(\mathbf_A (\omega)) = \operatorname(A)(1 - \operatorname(A)) ;
Covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
: \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 Kurt Friedrich Gödel ( , ; April 28, 1906 â€“ January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
described the ''representing function'' in his 1934 paper "On undecidable propositions of formal mathematical systems" (the "¬" indicates logical inversion, i.e. "NOT"):
Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
offers up the same definition in the context of the
primitive recursive function In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...
s 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 In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined ...
'', characteristic functions are generalized to take value in the real unit interval , or more generally, in some
algebra Algebra () is one of the 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 mathematics. Elementary a ...
or
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
(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 (mathematics), set. A poset consists of a set toget ...
or
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
). Such generalized characteristic functions are more usually called
membership function In mathematics, an indicator function or a characteristic function of a subset 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 ...
s, 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 mathematics ...
seen in many real-world
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
s like "tall", "warm", etc.


Derivatives of the indicator function

A particular indicator function is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
H(x) := \mathbf_ The
distributional derivative Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
of the Heaviside step function is equal to the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
.


See also

*
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
*
Laplacian of the indicator In mathematics, the Laplacian of the indicator of the domain ''D'' is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the ''surface'' of ''D''. It can be viewed as the ''surface delta pr ...
*
Dirac delta In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
* Extension (predicate logic) *
Free variables and bound variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
*
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
* Iverson bracket *
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
, a function that can be viewed as an indicator for the
identity relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
*
Macaulay brackets Macaulay brackets are a notation used to describe the ramp function :\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end A popular alternative transcription uses angle brackets, ''viz.'' \langle x \rangle.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 e ...
*
Membership function In mathematics, an indicator function or a characteristic function of a subset 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 ...
*
Simple function In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, ...
* Dummy variable (statistics) *
Statistical classification In statistics, classification is the problem of identifying which of a set of categories (sub-populations) an observation (or observations) belongs to. Examples are assigning a given email to the "spam" or "non-spam" class, and assigning a diagno ...
*
Zero-one loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...


Notes


References


Sources

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