Indicator Function

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## Definition

## Notation and terminology

## Basic properties

## Mean, variance and covariance

## Characteristic function in recursion theory, Gödel's and Kleene's representing function

## Characteristic function in fuzzy set theory

## Derivatives of the indicator function

## See also

## Notes

## References

## Sources

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Indicator Function

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In mathematics, an **indicator function** or a **characteristic function** of a subset A of a set X is a function defined from X to the two-element set , typically denoted as , and it indicates whether an element in X belongs to A; if an element in X belongs to A, and if does not belong to A. It is also denoted by to emphasize the fact that this function identifies the subset A of X.

In other contexts, such as computer science, this would more often be described as a **boolean predicate function** (to test set inclusion).

The Dirichlet function is an example of an indicator function and is the indicator of the rationals.

The indicator function of a subset A of a set X is a function

defined as

The Iverson bracket provides the equivalent notation, or to be used instead of .

The function is sometimes denoted , , K_{A} or even just .^{[a]}^{[b]}

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*^{[a]} 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 {0, 1}.

This mapping is surjective only when A is a non-empty proper subset of X. If A ? X, then **1**_{A} = 1. By a similar argument, if A ? ? then **1**_{A} = 0.

In the following, the dot represents multiplication, 1·1 = 1, 1·0 = 0 etc. "+" and "-" represent addition and subtraction. "" and "" is intersection and union, respectively.

If and are two subsets of , then

and the indicator function of the complement of i.e. is:

- .

More generally, suppose is a collection of subsets of X. For any :

is clearly a product of 0s and 1s. This product has the value 1 at precisely those that belong to none of the sets and is 0 otherwise. That is

Expanding the product on the left hand side,

where 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. The notation is used in other places as well, for instance in probability theory: if X is a probability space with probability measure and A is a measurable set, then 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. 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.)

Given a probability space with , the indicator random variable is defined by if otherwise

- Mean
- (also called "Fundamental Bridge").

Kurt Gödel described the *representing function* in his 1934 paper "On undecidable propositions of formal mathematical systems":^{[1]}

- "There shall correspond to each class or relation R a representing function ?(x
_{1}, ... x_{n}) = 0 if R(x_{1}, ... x_{n}) and ?(x_{1}, ... x_{n}) = 1 if ¬R(x_{1}, ... x_{n})."^{[1]}(the "¬" indicates logical inversion, i.e. "NOT")

Kleene (1952)^{[2]} offers up the same definition in the context of the primitive recursive functions as a function ? of a predicate P takes on values 0 if the predicate is true and 1 if the predicate is false.

For example, because the product of characteristic functions ?_{1}*?_{2}* ... *?_{n} = 0 whenever any one of the functions equals 0, it plays the role of logical OR: IF ?_{1} = 0 OR ?_{2} = 0 OR ... OR ?_{n} = 0 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 (p. 228), the bounded- (p. 228) and unbounded- (p. 279 ff) mu operators (Kleene (1952)) and the CASE function (p. 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.

A particular indicator function is the Heaviside step function. The Heaviside step function H(x) is the indicator function of the one-dimensional positive half-line, i.e. the domain [0, ?). The distributional derivative of the Heaviside step function is equal to the Dirac delta function, i.e.

with the following property:

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 *?*_{S}(* x*):

where *n* is the outward normal of the surface *S*. This 'surface delta function' has the following property:^{[3]}

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*.

- Dirac measure
- Laplacian of the indicator
- Dirac delta
- Extension (predicate logic)
- Free variables and bound variables
- Heaviside step function
- Iverson bracket
- Kronecker delta, a function that can be viewed as an indicator for the identity relation
- Macaulay brackets
- Multiset
- Membership function
- Simple function
- Dummy variable (statistics)
- Statistical classification
- Zero-one loss function

- ^
^{a}^{b}The Greek letter appears because it is the initial letter of the Greek word , 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 .

- ^
^{a}^{b}Davis, Martin, ed. (1965).*The Undecidable*. New York, NY: Raven Press Books. pp. 41-74. **^**Kleene, Stephen (1971) [1952].*Introduction to Metamathematics*(Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company. p. 227.**^**Lange, Rutger-Jan (2012). "Potential theory, path integrals and the Laplacian of the indicator".*Journal of High Energy Physics*.**2012**(11): 29-30. arXiv:1302.0864. Bibcode:2012JHEP...11..032L. doi:10.1007/JHEP11(2012)032. S2CID 56188533.

- Folland, G.B. (1999).
*Real Analysis: Modern Techniques and Their Applications*(Second ed.). John Wiley & Sons, Inc. ISBN 978-0-471-31716-6. - Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Section 5.2: Indicator random variables".
*Introduction to Algorithms*(Second ed.). MIT Press and McGraw-Hill. pp. 94-99. ISBN 978-0-262-03293-3. - 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. - Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002).
*Computability and Logic*. Cambridge UK: Cambridge University Press. ISBN 978-0-521-00758-0. - Zadeh, Lotfi A. (June 1965). "Fuzzy sets".
*Information and Control*.**8**(3): 338-353. doi:10.1016/S0019-9958(65)90241-X. - Goguen, Joseph (1967). "
*L*-fuzzy sets".*Journal of Mathematical Analysis and Applications*.**18**(1): 145-174. doi:10.1016/0022-247X(67)90189-8. hdl:10338.dmlcz/103980.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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