Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Indicator function
Summary
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 A is a subset of some set X, then if and otherwise, where is a common notation for the indicator function. Other common notations are and
The indicator function of A is the Iverson bracket of the property of belonging to A; that is,
For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers.
The indicator function of a subset A of a set X is a function
defined as
The Iverson bracket provides the equivalent notation, or ⟦x ∈ A⟧, to be used instead of
The function is sometimes denoted IA, χA, KA, or even just A.
The notation 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.
The indicator or characteristic function of a subset A of some set X maps elements of X to the range .
This mapping is surjective only when A is a non-empty proper subset of X. If then By a similar argument, if then
In the following, the dot represents multiplication, etc. "+" and "−" represent addition and subtraction.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (33)
Related people (2)
Related concepts (19)
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
Fuzzy set
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 a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
Measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral.
Related courses (6)
MATH-233: Probability and statistics
Le cours fournit une initiation à la théorie des probabilités et aux méthodes statistiques pour physiciens.
COM-417: Advanced probability and applications
In this course, various aspects of probability theory are considered. The first part is devoted to the main theorems in the field (law of large numbers, central limit theorem, concentration inequaliti
FIN-415: Probability and stochastic calculus
This course gives an introduction to probability theory and stochastic calculus in discrete and continuous time. We study fundamental notions and techniques necessary for applications in finance such