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[[Image:Indicator function illustration.png|right|thumb|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 (mathematics)|function]] defined on a [[Set (mathematics)|set]] ''X'' that indicates membership of an [[Element (mathematics)|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

:<math>\mathbf{1}_A \colon X \to \{ 0,1 \} \,</math>

defined as

:<math>\mathbf{1}_A(x) :=
\begin{cases}
1 &\text{if } x \in A, \\
0 &\text{if } x \notin A.
\end{cases}
</math>

The [[Iverson bracket]] allows the equivalent notation, [''x'' ∈ ''A''], to be used instead of '''1'''''A''(''x'').

The function '''1'''''A'' is sometimes denoted '''1'''''A'' ∈ A, χ''A'' or '''I'''''A'' or even just ''A''. (The [[Greek alphabet|Greek letter]] χ appears because it is the initial letter of the Greek word ''characteristic''.)

==Remark on notation and terminology==
* The notation '''1'''''A'' is also used to denote the [[identity function]].{{clarify|date=July 2012}}
* The notation χ''A'' is also used to denote the [[Characteristic function (convex analysis)|characteristic function]] in [[convex analysis]].{{clarify|date=July 2012}}

A related concept in [[statistics]] is that of a [[dummy variable (statistics)|dummy variable]] (this must not be confused with "dummy variables" as that term is usually used in mathematics, also called a [[free variables and bound variables|bound variable]]).

The term "[[characteristic function (probability theory)|characteristic function]]" has an unrelated meaning in [[probability theory]]. For this reason, [[List of probabilists|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 (mathematics)|function]] of a subset ''A'' of some set ''X'', [[Map (mathematics)|maps]] elements of ''X'' to the [[Range (mathematics)|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'' ≡ Ø then '''1'''''A'' = 0.

In the following, the dot represents multiplication, 1·1 = 1, 1·0 = 0 etc. "+" and "−" represent addition and subtraction. "<math>\cap </math>" and "<math>\cup </math>" is intersection and union, respectively.

If <math>A</math> and <math>B</math> are two subsets of <math>X</math>, then
:<math>\mathbf{1}_{A\cap B} = \min\{\mathbf{1}_A,\mathbf{1}_B\} = \mathbf{1}_A \cdot\mathbf{1}_B,</math>
:<math>\mathbf{1}_{A\cup B} = \max\{{\mathbf{1}_A,\mathbf{1}_B}\} = \mathbf{1}_A + \mathbf{1}_B - \mathbf{1}_A \cdot\mathbf{1}_B,</math>
and the indicator function of the [[Complement (set theory)|complement]] of <math>A</math> i.e. <math>A^C</math> is:
:<math>\mathbf{1}_{A^\complement} = 1-\mathbf{1}_A</math>.

More generally, suppose <math>A_1, \dotsc, A_n</math> is a collection of subsets of ''X''. For any
''x'' ∈ ''X'':

:<math> \prod_{k \in I} ( 1 - \mathbf{1}_{A_k}(x))</math>

is clearly a product of 0s and 1s. This product has the value 1 at
precisely those ''x'' ∈ ''X'' which belong to none of the sets ''Ak'' and
is 0 otherwise. That is

:<math> \prod_{k \in I} ( 1 - \mathbf{1}_{A_k}) = \mathbf{1}_{X - \bigcup_{k} A_k} = 1 - \mathbf{1}_{\bigcup_{k} A_k}.</math>

Expanding the product on the left hand side,

: <math> \mathbf{1}_{\bigcup_{k} A_k}= 1 - \sum_{F \subseteq \{1, 2, \dotsc, n\}} (-1)^{|F|} \mathbf{1}_{\bigcap_F A_k} = \sum_{\emptyset \neq F \subseteq \{1, 2, \dotsc, n\}} (-1)^{|F|+1} \mathbf{1}_{\bigcap_F A_k} </math>

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 <math>X</math> is a [[probability space]] with probability measure <math>\mathbb{P}</math> and <math>A</math> is a [[Measure (mathematics)|measurable set]], then <math>\mathbf{1}_A</math> becomes a [[random variable]] whose [[expected value]] is equal to the probability of <math>A</math>:

:<math>\operatorname{E}(\mathbf{1}_A)= \int_{X} \mathbf{1}_A(x)\,d\mathbb{P} = \int_{A} d\mathbb{P} = \operatorname{P}(A)</math>.

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]] <math>\textstyle (\Omega, \mathcal F, \mathbb P)</math> with <math>A \in \mathcal F</math>, the indicator random variable <math>\mathbf{1}_A \colon \Omega \rightarrow \Bbb{R}</math> is defined by <math>\mathbf{1}_A (\omega) = 1 </math> if <math> \omega \in A,</math> otherwise <math>\mathbf{1}_A (\omega) = 0.</math>

;[[Mean]]: <math>\operatorname{E}(\mathbf{1}_A (\omega)) = \operatorname{P}(A) </math>

;[[Variance]]: <math>\operatorname{Var}(\mathbf{1}_A (\omega)) = \operatorname{P}(A)(1 - \operatorname{P}(A)) </math>

;[[Covariance]]: <math> \operatorname{Cov}(\mathbf{1}_A (\omega), \mathbf{1}_B (\omega)) = \operatorname{P}(A \cap B) - \operatorname{P}(A)\operatorname{P}(B) </math>

==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 function]]s 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 φ1*φ2* . . . *φn = 0 whenever any one of the functions equals 0, it plays the role of logical OR: IF φ1 = 0 OR φ2 = 0 OR . . . OR φn = 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 operator]]s (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 [[universal algebra|algebra]] or [[structure (mathematical logic)|structure]] (usually required to be at least a [[partially ordered set|poset]] or [[lattice (order)|lattice]]). Such generalized characteristic functions are more usually called [[membership function (mathematics)|membership function]]s, and the corresponding "sets" are called ''fuzzy'' sets. Fuzzy sets model the gradual change in the membership [[degree of truth|degree]] seen in many real-world [[predicate (mathematics)|predicate]]s like "tall", "warm", etc.

==Derivatives of the indicator function==
A particular indicator function, which is very well known, is the [[Heaviside step function]]. The Heaviside step function is the indicator function of the one-dimensional positive half-line, i.e. the domain [0, ∞). It is well-known that the [[distributional derivative]] of the Heaviside step function, indicated by ''H''(''x''), is equal to the [[Dirac delta function]], i.e.

:<math>
\delta(x)=\tfrac{d H(x)}{dx},
</math>

with the following property:

:<math>
\int_{-\infty}^\infty f(x) \, \delta(x) dx = f(0).
</math>

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 [[Laplacian of the indicator#Surface Dirac delta function|inward normal derivative of the indicator]] gives rise to a 'surface delta function', which can be indicated by δ''S''('''x'''):

:<math>\delta_S(\mathbf{x})=-\mathbf{n}_x\cdot\nabla_x\mathbf{1}_{\mathbf{x}\in D}</math>

where ''n'' is the outward [[Normal (geometry)|normal]] of the surface ''S''. This 'surface delta function' has the following property:<ref>{{citation|last=Lange|first=Rutger-Jan|year=2012|publisher=Springer|title=Potential theory, path integrals and the Laplacian of the indicator|journal=Journal of High Energy Physics|volume=2012|pages=29–30|url=http://link.springer.com/article/10.1007%2FJHEP11(2012)032|issue=11|bibcode=2012JHEP...11..032L|doi=10.1007/JHEP11(2012)032|arxiv = 1302.0864 }}</ref>

:<math>
-\int_{\mathbf{R}^n}f(\mathbf{x})\,\mathbf{n}_x\cdot\nabla_x\mathbf{1}_{\mathbf{x}\in D}\;d^{n}\mathbf{x}=\oint_{S}\,f(\mathbf{\beta})\;d^{n-1}\mathbf{\beta}.
</math>

By setting the function ''f'' equal to one, it follows that the [[Laplacian of the indicator#Surface Dirac delta function|inward normal derivative of the indicator]] integrates to the numerical value of the [[surface area]] ''S''.

==See also==
* [[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 [[Equality (mathematics)|identity relation]]
* [[Multiset]]
* [[Membership function (mathematics)|Membership function]]
* [[Simple function]]
* [[Dummy variable (statistics)]]

{{More footnotes|date=December 2009}}

== Notes ==
{{Reflist}}

== References ==
* {{cite book | last = Folland | first = G.B. | title = Real Analysis: Modern Techniques and Their Applications | edition = Second | publisher = John Wiley & Sons, Inc. | year = 1999 }}
* {{cite book
| first1 = Thomas H. | last1 = Cormen | authorlink1 = Thomas H. Cormen
| first2 = Charles E. | last2 = Leiserson | authorlink2 = Charles E. Leiserson
| first3 = Ronald L. | last3 = Rivest | authorlink3 = Ronald L. Rivest
| first4 = Clifford | last4 = Stein | authorlink4 = Clifford Stein
| title = [[Introduction to Algorithms]]
| edition = Second Edition
| publisher = MIT Press and McGraw-Hill
| year = 2001
| isbn = 0-262-03293-7
| chapter = Section 5.2: Indicator random variables
| pages = 94–99
}}
* {{cite book
| editor1-first = Martin | editor1-last = Davis | editor1-link = Martin Davis
| year = 1965
| title = The Undecidable
| publisher = Raven Press Books, Ltd. | location = New York
}}
* {{cite book
| first = Stephen | last = Kleene | authorlink = Stephen Kleene
| origyear = 1952 | title = Introduction to Metamathematics
| publisher = Wolters-Noordhoff Publishing and North Holland Publishing Company | location = Netherlands
| type = Sixth Reprint with corrections
| year = 1971
}}
* {{cite book
| first1 = George | last1 = Boolos | authorlink1 = George Boolos
| first2 = John P. | last2 = Burgess | authorlink2 = John P. Burgess
| first3 = Richard C. | last3 = Jeffrey | authorlink3 = Richard C. Jeffrey
| year = 2002
| title = Computability and Logic
| publisher = Cambridge University Press | location = Cambridge UK
| isbn = 0-521-00758-5
}}
* {{cite journal
| first = Lotfi A. | last = Zadeh | authorlink = Lotfi A. Zadeh
| date = June 1965
| title = Fuzzy sets
| journal = [[Information and Control]]
| volume = 8 | issue = 3 | pages = 338–353
| url = http://www-bisc.cs.berkeley.edu/zadeh/papers/Fuzzy%20Sets-1965.pdf | format = PDF
|
doi = 10.1016/S0019-9958(65)90241-X
}}
* {{cite journal
| first = Joseph | last = Goguen | authorlink = Joseph Goguen
| year = 1967
| title = ''L''-fuzzy sets
| journal = Journal of Mathematical Analysis and Applications
| volume = 18 | issue = 1 | pages = 145–174
| doi = 10.1016/0022-247X(67)90189-8
}}

[[Category:Measure theory]]
[[Category:Integral calculus]]
[[Category:Real analysis]]
[[Category:Mathematical logic]]
[[Category:Basic concepts in set theory]]
[[Category:Probability theory]]
[[Category:Types of functions]]

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