indicator function
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The graph of the indicator function of a two-dimensional subset of a square.
In
mathematics, an
indicator function or a
characteristic function is a
function defined on a
set X
that indicates membership of an element in a
subset A
of
X
, having the value 1 for all elements of
A and the value 0 for all elements of
X not in
A.
Definition
The indicator function of a subset
A
of a set
X
is a function
defined as
begin{cases} 1 &mbox{if} x in A, end{cases}The
Iverson bracket allows the equivalent notation,
[x ∈ A]
, to be used instead of
1A(x) .
The indicator function of
A
is sometimes denoted
χA(x)
or
IA(x)
or even
A(x)
.
(The
Greek letter χ because it is the initial letter of the Greek
etymon of the word
characteristic.)
Remark on notation and terminology
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
probability theory. For this reason,
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 which indicates membership in a set.
Basic properties
The
indicator or
characteristic function of a subset
A
of some set
X
, maps elements of
A
to the range
01
.This mapping is
surjective only when
A
is a
proper subset of
X
. If
A ≡ X
, then
1A = 1
. By a similar argument, if
A ≡ &(hi;
then
1A = 0
.In the following, the dot represents multiplication, 1·1 = 1, 1·0 = 0 etc. "+" and "−" represent addition and subtraction. "
&ca(;
" and "
&cu(;
" is intersection and union, respectively. If
A
and
B
are two subsets of
X
, then
1A&ca(; B = m∈1A1B = 1A cderiv(⋅)1B
1A&cu(; B = max1A1B = 1A + 1B - 1A cderiv(⋅)1B
and the "complement" of the indicator function of A i.e. A
C is:
1Aco&(lusmn;≤ment = 1-1A.
More generally, suppose
A1 lderiv(⋅)s An
is a collection of subsets of
X
. For any
x ∈ X
,
&(rod;k ∈ I ( 1 - 1Ak(x))
is clearly a product of
0
s and
1
s. This product has the value 1 atprecisely those
x ∈ X
which belong to none of the sets
Ak
andis
0
otherwise. That is
&(rod;k ∈ I ( 1 - 1Ak) = 1X - big&cu(;k Ak = 1 - 1big&cu(;k Ak.
Expanding the product on the left hand side,
1big&cu(;k Ak= 1 - ΣF &su(;eq 1 2 lderiv(⋅)s n (-1)||F|| 1big&ca(;F Ak = Σe&(lusmn;tyset ≠q F &su(;eq 1 2 lderiv(⋅)s n (-1)||F||+1 1big&ca(;F Ak
where
||F||
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
P
and
A
is a
measurable set, then
1A
becomes a
random variable whose
expected value is equal to the probability of
A:
E(1A)= ∈tX 1A(x)dP = ∈tA dP = P(A). &nbs(;&nbs(;
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.)
Mean, Variance and covariance
Given a
probability space:
textsty≤ (Ω Scri(t F P)
with A
∈ Scri(t F
, the indicator random variable
1A : Ω → BbbR
, is defined by
1A (ω) = 1
if
ω ∈ A
otherwise
1A (ω) = 0
(i.e basically
1A
is a indicator random variable)
Cov(1A (ω) 1B (ω)) = P(A &ca(; B) - P(A)P(B)
(
Covariance)
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 paper appears on pp. 41-74 in
Martin Davis ed.
The Undecidable):
"There shall correspond to each class or relation R a representing function φ(x1, . . ., xn) = 0 if R(x1, . . ., xn) and φ(x1, . . ., xn)=1 if ~R(x1, . . ., xn)." (p. 42; the "~" indicates logical inversion i.e. "NOT")
Stephen Kleene (1952) (p. 227) 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.For example, because the product of characteristic functions φ
1*φ
2* . . . *φ
n = 0 whenever any one of the functions equals 0, it plays the role of logical OR: IF φ
1=0 OR φ
2=0 OR . . . OR φ
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. 279ff)
mu operators (Kleene (1952)) and the CASE function (p. 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.
See also
{{More footnotes|date=December 2009}}
References
- Folland, G.B.; Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 5.2: Indicator random variables, pp. 94–99.
- Martin Davis ed. (1965), The Undecidable, Raven Press Books, Ltd., New York.
- Stephen Kleene, (1952), Introduction to Metamathematics, Wolters-Noordhoff Publishing and North Holland Publishing Company, Netherlands, Sixth Reprint with corrections 1971.
- George Boolos, John P. Burgess, Richard C. Jeffrey (2002), Computability and Logic, Cambridge University Press, Cambridge UK, ISBN 0-521-00758-5.
- Lotfi A. Zadeh, 1965, "Fuzzy sets". Information and Control 8: 338–353. weblink
- Joseph Goguen, 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
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