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,
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[a] to describe the function that indicates membership in a set.
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 with the indicator random variable is defined by if otherwise
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"):[1]: 42
There shall correspond to each class or relation R a representing function if and if
Kleene 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.[2]
For example, because the product of characteristic functions whenever any one of the functions equals 0, it plays the role of logical OR: IF OR OR ... OR 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,[2]: 228 the bounded-[2]: 228 and unbounded-[2]: 279 ff mu operators and the CASE function.[2]: 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.
In general, the indicator function of a set is not smooth; it is continuous if and only if its
support is a
connected component. In the
algebraic geometry of
finite fields, however, every
affine variety admits a (
Zariski) continuous indicator function.[3] Given a
finite set of functions let be their vanishing locus. Then, the function acts as an indicator function for . If then , otherwise, for some , we have , which implies that , hence .
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 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 :
where n is the outward
normal of the surface S. This 'surface delta function' has the following property:[4]
^
abThe
Greek letterχ appears because it is the initial letter of the Greek word χαρακτήρ, which is the ultimate origin of the word characteristic.
^The set of all indicator functions on X can be identified with the
power set of X. Consequently, both sets are sometimes denoted by This is a special case () of the notation for the set of all functions
References
^Davis, Martin, ed. (1965). The Undecidable. New York, NY: Raven Press Books. pp. 41–74.
^
abcdeKleene, Stephen (1971) [1952]. Introduction to Metamathematics (Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company. p. 227.
Davis, Martin, ed. (1965). The Undecidable. New York, NY: Raven Press Books.
Kleene, Stephen (1971) [1952]. Introduction to Metamathematics (Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company.