In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an indicator function or a characteristic function of a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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 , one has
if
and
otherwise, where
is a common notation for the indicator function. Other common notations are
and
The indicator function of is the
Iverson bracket
In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. ...
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
The indicator function of a subset of a set is a function
defined as
The
Iverson bracket
In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. ...
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: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
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
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
and
modern many-valued logic, predicates are the
characteristic functions of a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. 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
range \.
This mapping is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
only when is a non-empty
proper subset of . If
A \equiv X, then
\mathbf_A=1. By a similar argument, if
A\equiv\emptyset then
\mathbf_A=0.
In the following, the dot represents multiplication,
1\cdot1 = 1, 1\cdot0 = 0, etc. "+" and "−" represent addition and subtraction. "
\cap " and "
\cup " are intersection and union, respectively.
If
A and
B are two subsets of
X, then
\begin
\mathbf_ = \min\ = \mathbf_A \cdot\mathbf_B, \\
\mathbf_ = \max\ = \mathbf_A + \mathbf_B - \mathbf_A \cdot\mathbf_B,
\end
and the indicator function of the
complement of
A i.e.
A^C 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_ ( 1 - \mathbf_(x))
is clearly a product of s and s. This product has the value 1 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
. The notation is used in other places as well, for instance in
probability theory: if is a
probability space with probability measure
\operatorname and is a
measurable set, then
\mathbf_A becomes a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
whose
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
is equal to the probability of :
\operatorname(\mathbf_A)= \int_ \mathbf_A(x)\,d\operatorname = \int_ d\operatorname = \operatorname(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 In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural constructi ...
, 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 \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(\mathbf_A (\omega)) = \operatorname(A) (also called "Fundamental Bridge").
;
Variance:
\operatorname(\mathbf_A (\omega)) = \operatorname(A)(1 - \operatorname(A))
;
Covariance:
\operatorname(\mathbf_A (\omega), \mathbf_B (\omega)) = \operatorname(A \cap B) - \operatorname(A)\operatorname(B)
Characteristic function in recursion theory, Gödel's and Kleene's representing function
Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
described the ''representing function'' in his 1934 paper "On undecidable propositions of formal mathematical systems" (the "¬" 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
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
(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
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
seen in many real-world predicates like "tall", "warm", etc.
Derivatives of the indicator function
A particular indicator function is the Heaviside step function
H(x) := \mathbf_
The distributional derivative of the Heaviside step function is equal to the Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, i.e.
\frac=\delta(x)
and similarly the distributional derivative of G(x) := \mathbf_ 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\mathbf_
where is the outward normal of the surface . This 'surface delta function' has the following property:
-\int_f(\mathbf)\,\mathbf_x\cdot\nabla_x\mathbf_\;d^\mathbf = \oint_\,f(\mathbf)\;d^\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
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
.
See also
* Dirac measure
* Laplacian of the indicator
* Dirac delta
* Extension (predicate logic)
The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.
Examples
For example, the statement "''d2'' is the weekday following ''d1''" can ...
* Free variables and bound variables
* Heaviside step function
* Iverson bracket
In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. ...
* 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)
In regression analysis, a dummy variable (also known as indicator variable or just dummy) is one that takes the values 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome. For example, i ...
* Statistical classification
* Zero-one loss function
Notes
References
Sources
*
*
*
*
*
*
*
{{refend
Measure theory
Integral calculus
Real analysis
Mathematical logic
Basic concepts in set theory
Probability theory
Types of functions