In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 is a subset of some set , then the indicator function of is the function
defined by
if
and
otherwise. Other common notations are and
The indicator function of is the
Iverson bracket of the property of belonging to ; that is,
For example, the
Dirichlet function is the indicator function of the
rational numbers as a subset of the
real numbers.
Definition
Given an arbitrary set , the indicator function of a subset of is the function
defined by
The
Iverson bracket provides the equivalent notation
Notation and terminology
The notation
\chi_A 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
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
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.
Basic properties
The ''indicator'' or ''characteristic''
function of a subset of some set
maps elements of to the
codomain \.
This mapping is
surjective only when is a non-empty
proper subset of . If
A = X, then
\mathbf_A \equiv 1. By a similar argument, if
A = \emptyset then
\mathbf_A \equiv 0.
If
A and
B are two subsets of
X, then
\begin
\mathbf_(x) ~&=~ \min\bigl\ ~~=~ \mathbf_A(x) \cdot\mathbf_B(x), \\
\mathbf_(x) ~&=~ \max\bigl\ ~=~ \mathbf_A(x) + \mathbf_B(x) - \mathbf_A(x) \cdot \mathbf_B(x)\,,
\end
and the indicator function of the
complement of
A i.e.
A^\complement is:
\mathbf_ = 1 - \mathbf_A.
More generally, suppose
A_1, \dotsc, A_n is a collection of subsets of . For any
x \in X:
\prod_ \left(\ 1 - \mathbf_\!\left( x \right)\ \right)
is a product of s and s. This product has the value at precisely those
x \in X that belong to none of the sets
A_k and is 0 otherwise. That is
\prod_ ( 1 - \mathbf_) = \mathbf_ = 1 - \mathbf_.
Expanding the product on the left hand side,
\mathbf_= 1 - \sum_ (-1)^ \mathbf_ = \sum_ (-1)^ \mathbf_
where
, F, is the
cardinality of . 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
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
: if is a
probability space with probability measure
\mathbb and is a
measurable set, then
\mathbf_A becomes a
random variable whose
expected value is equal to the probability of :
\operatorname\mathbb_X\left\\ =\ \int_ \mathbf_A( x )\ \operatorname(x) = \int_ \operatorname(x) = \operatorname\mathbb(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
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the
Möbius function. (See paragraph below about the use of the inverse in classical recursion theory.)
Mean, variance and covariance
Given a
probability space \textstyle (\Omega, \mathcal F, \operatorname) with
A \in \mathcal F, the indicator random variable
\mathbf_A \colon \Omega \rightarrow \mathbb is defined by
\mathbf_A (\omega) = 1 if
\omega \in A, otherwise
\mathbf_A (\omega) = 0.
;
Mean:
\ \operatorname\mathbb(\mathbf_A (\omega)) = \operatorname\mathbb(A)\ (also called "Fundamental Bridge").
;
Variance:
\ \operatorname(\mathbf_A (\omega)) = \operatorname\mathbb(A)(1 - \operatorname\mathbb(A)).
;
Covariance:
\ \operatorname(\mathbf_A (\omega), \mathbf_B (\omega)) = \operatorname\mathbb(A \cap B) - \operatorname\mathbb(A) \operatorname\mathbb(B).
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 symbol "" indicates logical inversion, i.e. "NOT"):
Kleene offers up the same definition in the context of the
primitive recursive functions as a function of a predicate takes on values if the predicate is true and if the predicate is false.
For example, because the product of characteristic functions
\phi_1 * \phi_2 * \cdots * \phi_n = 0 whenever any one of the functions equals , it plays the role of logical OR: IF
\phi_1 = 0\ OR
\ \phi_2 = 0 OR ... OR
\phi_n = 0 THEN their product is . What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is when the function is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY,
[ the bounded-][ and unbounded-][ mu operators and the CASE function.][
]
Characteristic function in fuzzy set theory
In classical mathematics, characteristic functions of sets only take values (members) or (non-members). In '' fuzzy set theory'', characteristic functions are generalized to take value in the real unit interval , 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.
Smoothness
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
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
of finite fields, however, every affine variety admits a ( Zariski) continuous indicator function. Given a finite set of functions f_\alpha \in \mathbb_q\left x_1, \ldots, x_n\right/math> let V = \bigl\ be their vanishing locus. Then, the function \mathbb(x) = \prod\left(\ 1 - f_\alpha(x)^\right) acts as an indicator function for V. If x \in V then \mathbb(x) = 1, otherwise, for some f_\alpha, we have f_\alpha(x) \neq 0 which implies that f_\alpha(x)^ = 1, hence \mathbb(x) = 0.
Although indicator functions are not smooth, they admit weak derivatives. For example, consider Heaviside step function H(x) \equiv \operatorname\mathbb\!\bigl(x > 0\bigr) The distributional derivative of the Heaviside step function is equal to the Dirac delta function, i.e. \frac= \delta(x)
and similarly the distributional derivative of G(x) := \operatorname\mathbb\!\bigl(x < 0\bigr) is \frac = -\delta(x).
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 . The surface of will be denoted by . 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 \delta_S(\mathbf):
\delta_S(\mathbf) = -\mathbf_x \cdot \nabla_x \operatorname\mathbb\!\bigl(\ \mathbf\in D\ \bigr)\
where is the outward normal of the surface . This 'surface delta function' has the following property:
-\int_f(\mathbf)\,\mathbf_x\cdot\nabla_x \operatorname\mathbb\!\bigl(\ \mathbf\in D\ \bigr) \; \operatorname^\mathbf = \oint_\,f(\mathbf) \; \operatorname^\mathbf.
By setting the function equal to one, it follows that the inward normal derivative of the indicator integrates to the numerical value of the surface area .
See also
* Dirac measure
* Laplacian of the indicator
* Dirac delta
* Extension (predicate logic)
* Free variables and bound variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...
* Heaviside step function
* Identity 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
* Subobject classifier, a related concept from topos theory.
Notes
References
Sources
*
*
*
*
*
*
*
{{refend
Measure theory
Integral calculus
Real analysis
Mathematical logic
Basic concepts in set theory
Probability theory
Types of functions