Indicator function
The indicator function of A is the Iverson bracket of the property of belonging to A; that is,
The Iverson bracket provides the equivalent 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.
The term "characteristic function" has an unrelated meaning in classic probability theory.
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 codomain
This mapping is surjective only when A is a non-empty proper subset of X .
Expanding the product on the left hand side,
is the cardinality of F. 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.
becomes a random variable whose expected value is equal to the probability of A:
This identity is used in a simple proof of Markov's inequality.
In many cases, such as order theory, the inverse of the indicator function may be defined.
(See paragraph below about the use of the inverse in classical recursion theory.)
Kurt Gödel described the representing function in his 1934 paper "On undecidable propositions of formal mathematical systems" (the symbol "¬" indicates logical inversion, i.e. "NOT"):[1]: 42 There shall correspond to each class or relation R a representing function
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.
whenever any one of the functions equals 0, it plays the role of logical OR: IF
[2]: 229 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.
Although indicator functions are not smooth, they admit weak derivatives.
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]
By setting the function f equal to one, it follows that the inward normal derivative of the indicator integrates to the numerical value of the surface area S.
