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{{Short description|Mathematical function characterizing set membership}}
{{About|the 0
{{More footnotes|date=December 2009}}
{{Use American English|date = March 2019}}
[[Image:Indicator function illustration.png|right|thumb|A three-dimensional plot of an indicator function, shown over a square two-dimensional ___domain (set {{mvar|X}}): the "raised" portion overlays those two-dimensional points which are members of the "indicated" subset ({{mvar|A}}).]]
In [[mathematics]], an '''indicator function''' or a '''characteristic function''' of a [[subset]] of a [[Set (mathematics)|set]] is a [[Function (mathematics)|function]] that maps elements of the subset to one, and all other elements to zero. That is, if {{mvar|A}} is a subset of some set {{mvar|X}},
The indicator function
For example, the [[Dirichlet function]] is the indicator function of the [[rational number]]s as a subset of the [[real number]]s.
==Definition==
▲<math display=block>\mathbf{1}_A \colon X \to \{ 0, 1 \} </math>
▲defined as
▲<math display=block>\mathbf{1}_A(x) :=
\begin{cases}
1
0
\end{cases}
</math>
The [[Iverson bracket]] provides the equivalent notation
The function <math>\mathbf{1}_A</math> is sometimes denoted {{
The [[Greek alphabet|Greek letter]] {{mvar|χ}} appears because it is the initial letter of the Greek word {{lang|grc|{{math|χαρακτήρ}}}}, which is the ultimate origin of the word ''characteristic''. }} or even just {{mvar|A}}.{{efn| The set of all indicator functions on {{mvar|X}} can be identified with the set operator <math>\mathcal{P}(X),</math> the [[power set]] of {{mvar|X}}. Consequently, both sets are }} ==Notation and terminology==
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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|classic probability theory]]. For this reason, [[List of probabilists|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''
In [[fuzzy logic]] and [[Many-valued logic|modern many-valued logic]], predicates are the [[characteristic function (probability theory)|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 (mathematics)|function]] of a subset {{mvar|A}} of some set {{mvar|X}} [[Map (mathematics)|maps]] elements of {{mvar|X}} to the [[
This mapping is [[surjective]] only when {{mvar|A}} is a non-empty [[proper subset]] of {{mvar|X}}. If <math>A
If <math>A</math> and <math>B</math> are two subsets of <math>X,</math> then
<math display=block>\begin{align}
\mathbf{1}_{A\cap B}(x) ~&=~ \min\bigl\{\mathbf{1}_A(x),\ \mathbf{1}_B(x)\bigr\} ~~=~ \mathbf{1}_A(x) \cdot\mathbf{1}_B(x), \\
\mathbf{1}_{A\cup B}(x) ~&=~ \max\
\end{align}</math>
and the indicator function of the [[Complement (set theory)|complement]] of <math>A</math> i.e. <math>A^
<math display=block>\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 {{mvar|X}}. For any <math>x \in X:</math>
<math display=block> \prod_{k \in I} \left(\ 1 - \mathbf{1}_{A_k}\!\left( x \right)\ \right)</math>
is
<math display=block> \prod_{k \in I} ( 1 - \mathbf{1}_{A_k}) = \mathbf{1}_{X - \bigcup_{k} A_k} = 1 - \mathbf{1}_{\bigcup_{k} A_k}.</math>
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Expanding the product on the left hand side,
<math display=block>
where <math>|F|</math> is the [[cardinality]] of {{mvar|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 {{mvar|X}} is a [[probability space]] with probability measure <math>\
<math display=block>\operatorname\mathbb{E}
This identity is used in a simple proof of [[Markov's inequality]].
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Given a [[probability space]] <math>\textstyle (\Omega, \mathcal F, \operatorname{P})</math> with <math>A \in \mathcal F,</math> the indicator random variable <math>\mathbf{1}_A \colon \Omega \rightarrow \mathbb{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\mathbb{E}(\mathbf{1}_A (\omega)) = \operatorname\mathbb{P}(A)\ </math>
;[[Variance]]: <math>\ \operatorname{Var}(\mathbf{1}_A (\omega)) = \operatorname\mathbb{P}(A)(1 - \operatorname\mathbb{P}(A))
;[[Covariance]]: <math>\ \operatorname{Cov}(\mathbf{1}_A (\omega), \mathbf{1}_B (\omega)) = \operatorname\mathbb{P}(A \cap B) - \operatorname\mathbb{P}(A) \operatorname\mathbb{P}(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 "{{math|¬}}" indicates logical inversion, i.e. "NOT"):<ref name=Martin-1965>{{cite book |pages=41–74 |editor-link=Martin Davis (mathematician) |editor-first=Martin |editor-last=Davis |year=1965 |title=The Undecidable |publisher=Raven Press Books |place=New York, NY}}</ref>{{rp|page=42}}
{{blockquote|1=There shall correspond to each class or relation {{mvar|R}} a representing function <math>\phi(x_1, \ldots x_n) = 0</math> if <math>R(x_1,\ldots x_n)</math> and <math>\phi(x_1,\ldots x_n) = 1</math> if <math>\neg R(x_1,\ldots x_n).</math>}}
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[[Stephen Kleene|Kleene]] offers up the same definition in the context of the [[primitive recursive function]]s as a function {{mvar|φ}} of a predicate {{mvar|P}} takes on values {{math|0}} if the predicate is true and {{math|1}} if the predicate is false.<ref name=Kleene1952>{{cite book |last=Kleene |first=Stephen |author-link=Stephen Kleene |year=1971 |orig-year=1952 |title=Introduction to Metamathematics |page=227 |publisher=Wolters-Noordhoff Publishing and North Holland Publishing Company |___location=Netherlands |edition=Sixth reprint, with corrections}}</ref>
For example, because the product of characteristic functions <math>\phi_1 * \phi_2 * \cdots * \phi_n = 0</math> whenever any one of the functions equals {{math|0}}, it plays the role of logical OR: IF <math>\phi_1 = 0\ </math> OR <math>\ \phi_2 = 0</math> OR
==Characteristic function in fuzzy set theory==
In classical mathematics, characteristic functions of sets only take values {{math|1}} (members) or {{math|0}} (non-members). In ''[[fuzzy set theory]]'', characteristic functions are generalized to take value in the real unit interval {{closed-closed|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.
==Smoothness==
{{
In general, the indicator function of a set is not smooth; it is continuous if and only if its [[support (math)|support]] is a [[connected component (topology)|connected component]]. In the [[algebraic geometry]] of [[finite fields]], however, every [[affine variety]] admits a ([[Zariski topology|Zariski]]) continuous indicator function.<ref>{{Cite book|title=Course in Arithmetic|last=Serre|pages=5}}</ref> Given a [[finite set]] of functions <math>f_\alpha \in \mathbb{F}_q\left[\ x_1, \ldots, x_n\right]</math> let <math>V = \bigl\{\ x \in \mathbb{F}_q^n : f_\alpha(x) = 0\ \bigr\}</math> be their vanishing locus. Then, the function <math display="inline">\mathbb{P}(x) = \prod\left(\ 1 - f_\alpha(x)^{q-1}\right)</math> acts as an indicator function for <math>V.</math> If <math>x \in V</math> then <math>\mathbb{P}(x) = 1,</math> otherwise, for some <math>f_\alpha,</math> we have <math>f_\alpha(x) \neq 0</math> which implies that <math>f_\alpha(x)^{q-1} = 1,</math> hence <math>\mathbb{P}(x) = 0.</math>
Although indicator functions are not smooth, they admit [[weak derivative]]s. For example, consider [[Heaviside step function]] <math display="block">H(x) \equiv \operatorname\mathbb{I}\!\bigl(x > 0\bigr)</math> The [[distributional derivative]] of the Heaviside step function is equal to the [[Dirac delta function]], i.e. <math display=block>\frac{\mathrm{d}H(x)}{\mathrm{d}x}= \delta(x)</math>
and similarly the distributional derivative of <math display="block">G(x) := \operatorname\mathbb{I}\!\bigl(x < 0\bigr)</math> is <math display=block>\frac{\mathrm{d}G(x)}{\mathrm{d}x} = -\delta(x).</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 {{mvar|D}}. The surface of {{mvar|D}} will be denoted by {{mvar|S}}. Proceeding, it can be derived that the [[Laplacian of the indicator#Dirac surface delta function|inward normal derivative of the indicator]] gives rise to a 'surface delta function', which can be indicated by <math>\delta_S(\mathbf{x}):</math>▼
<math display=block>\delta_S(\mathbf{x}) = -\mathbf{n}_x \cdot \nabla_x\mathbf{1}_{\mathbf{x}\in D}</math>▼
▲
▲<math display=block>\delta_S(\mathbf{x}) = -\mathbf{n}_x \cdot \nabla_x \
where {{mvar|n}} is the outward [[Normal (geometry)|normal]] of the surface {{mvar|S}}. This 'surface delta function' has the following property:<ref>{{cite journal |last=Lange |first=Rutger-Jan |year=2012 |title=Potential theory, path integrals and the Laplacian of the indicator |journal=Journal of High Energy Physics |volume=2012 |issue=11 |pages=29–30 |arxiv=1302.0864 |bibcode=2012JHEP...11..032L |doi=10.1007/JHEP11(2012)032|s2cid=56188533 }}</ref>
<math display=block>-\int_{\R^n}f(\mathbf{x})\,\mathbf{n}_x\cdot\nabla_x \
▲<math display=block>-\int_{\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 {{mvar|f}} equal to one, it follows that the [[Laplacian of the indicator#Dirac surface delta function|inward normal derivative of the indicator]] integrates to the numerical value of the [[surface area]] {{mvar|S}}.
==See also==
{{Div col|colwidth=
* [[Dirac measure]]
* [[Laplacian of the indicator]]
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* [[Free variables and bound variables]]
* [[Heaviside step function]]
* [[Identity function]]
* [[Iverson bracket]]
* [[Kronecker delta]], a function that can be viewed as an indicator for the [[Equality (mathematics)|identity relation]]
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* [[Statistical classification]]
* [[Zero-one loss function]]
*[[Subobject classifier]], a related concept from [[Topos theory|topos theory]].{{div col end}}
==Notes==
{{notelist
==References==
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==Sources==
{{refbegin|
* {{cite book |last=Folland |first=G.B. |title=Real Analysis: Modern Techniques and Their Applications |publisher=John Wiley & Sons, Inc. |year=1999 |isbn=978-0-471-31716-6 |edition=Second}}
* {{cite book |last1=Cormen |first1=Thomas H. |title=Introduction to Algorithms |title-link=Introduction to Algorithms |last2=Leiserson |first2=Charles E. |last3=Rivest |first3=Ronald L. |last4=Stein |first4=Clifford |publisher=MIT Press and McGraw-Hill |year=2001 |isbn=978-0-262-03293-3 |edition=Second |pages=[https://archive.org/details/introductiontoal00corm_691/page/n116 94]–99 |chapter=Section 5.2: Indicator random variables |author-link=Thomas H. Cormen |author-link2=Charles E. Leiserson |author-link3=Ronald L. Rivest |author-link4=Clifford Stein}}
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