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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
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