Indicator function: Difference between revisions

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{{shortShort description|TypeMathematical offunction mathematicalcharacterizing functionset membership}}

{{About|the 0-–1 indicator function|the 0-–infinity indicator function|characteristic function (convex analysis)}}

{{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]] {{mvar|A}} of a [[Set (mathematics)|set]] {{mvar|X}} is a [[Function (mathematics)|function]] definedthat frommaps {{mvar|X}}elements toof the two-elementsubset setto <math>\{ 0one, 1and \}</math>,all typicallyother denotedelements asto <math>\mathbf{1}_Azero. \colonThat Xis, \toif \{ 0, 1 \{mvar|A}} </math>,is anda itsubset indicatesof whethersome an [[Element (mathematics)|element]] inset {{mvar|X}}, belongsthen tothe indicator function of {{mvar|A}}; is the function <math>\mathbf{1}_A</math> defined by <math>\mathbf{1}_{A}\!(x) = 1</math> if an element <math>x</math> \in {{mvar|X}} belongs to {{mvar|A}},</math> and <math>\mathbf{1}_{A}\!(x) = 0</math> ifotherwise. <math>x</math>Other doescommon notnotations belong toare {{mvarmath|𝟙{{sub|''A''}}}}. It is also denoted byand <math>I_A\chi_A.</math> to emphasize the fact that this function identifies the [[subset]] {{mvarefn|Aname=χαρακτήρ}} of {{mvar|X}}.

The indicator function of {{mvar|A}} is the [[Iverson bracket]] of the property of belonging to {{mvar|A}}; that is,

In other contexts, such as [[computer science]], this would more often be described as a '''boolean [[Predicate (mathematical logic)|predicate]] function''' (to test set inclusion).

<math display="block">\mathbf{1}_{A}(x) = \left[\ x\in A\ \right].</math>

The [[Dirichlet function]] is an example of an indicator function and is the indicator of the [[rationals]].

For example, the [[Dirichlet function]] is the indicator function of the [[rational number]]s as a subset of the [[real number]]s.

==Definition==

TheGiven an arbitrary set {{mvar|X}}, the indicator function of a subset {{mvar|A}} of a set {{mvar|X}} is athe function

:<math display=block>\mathbf{1}_A \colon X \tomapsto \{ 0, 1 \} </math>

defined asby

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

<math display="block" qid="Q371983">\operatorname\mathbf{1}_A\!( x ) =

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]] provides the equivalent notation, <math>\left[\ x\in A\ \right]</math> or {{nowrapnobr|{{math|⧙ ⟦&thinsp;''x'' ϵ∈ ''A'' ⧘&thinsp;⟧}},}} tothat can be used instead of <math>\mathbf{1}_{A}\!(x).</math>.

The function <math>\mathbf{1}_A</math> is sometimes denoted <{{math>I_A</math>|𝟙{{sub|''A''}}}}, {{mvar|I<mathsub>\chi_AA</mathsub>}}, {{mvar|K}}&chi;_{{mvar|A}} or even just <math>A</math>.}}{{efn|name=χαρακτήρ|
The [[Greek alphabet|Greek letter]] <math>\{{mvar|&chi</math>;}} 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 sometimes denoted by the conventional [[abuse of notation]] as <math>2^X,</math> in analogy to the relation for the count of elements in the powerset and the original set. This is a special case (<math>\left(Y = \{0,\, 1\} = 2\right)</math>) of the notation <math>Y^X</math> for the set of all functions <math>f</math> such that <math>f: X \tomapsto Y \,.</math>.
}}

==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''{{efn|name=χαρακτήρ}} to describe the function that indicates membership in a set.

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 [[Range of a function|rangecodomain]] <math>\{0,\, 1\}.</math>

This mapping is [[surjective]] only when {{mvar|A}} is a non-empty [[proper subset]] of {{mvar|X}}. If {{mvar|<math>A}} ≡= {{mvar|X}},</math> then '''1'''<submath>\mathbf{{mvar|A1}}</sub>_A =\equiv 1.</math> By a similar argument, if {{mvar|<math>A}} ≡= ∅\emptyset</math> then '''1'''<submath>\mathbf{{mvar|A1}}</sub>_A =\equiv 0.</math>

If <math>A</math> and <math>B</math> are two subsets of <math>X,</math> then

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.

<math display=block>\begin{align}

:<math>\mathbf{1}_{A\cupcap B}(x) ~&=~ \maxmin\bigl\{{\mathbf{1}_A(x),\ \mathbf{1}_B}(x)\bigr\} ~~=~ \mathbf{1}_A +(x) \cdot\mathbf{1}_B -(x), \mathbf{1}_A \cdot\mathbf{1}_B,</math>

\mathbf{1}_{A\cup B}(x) ~&=~ \max\bigl\{\mathbf{1}_A(x),\ \mathbf{1}_B(x)\bigr\} ~=~ \mathbf{1}_A(x) + \mathbf{1}_B(x) - \mathbf{1}_A(x) \cdot \mathbf{1}_B(x)\,,

\end{align}</math>

Ifand <math>A</math>the andindicator function of the [[Complement (set theory)|complement]] of <math>BA</math> are two subsets ofi.e. <math>XA^\complement</math>, thenis:

:<math display=block>\mathbf{1}_{A^\cap Bcomplement} = \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 {{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 clearly a product of 0s{{math|0}}s and 1s{{math|1}}s. This product has the value {{math|1}} at precisely those <math>x \in X</math> that belong to none of the sets <math>A_k</math> and is 0 otherwise. That 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>

Expanding the product on the left hand side,

: <math> display=block>\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 <math>|F|</math> is the [[cardinality]] of {{mvar|F}}{{explain|{{mvar|F}} is defined if we know what the integer n is. But what is it supposed to represent here? Are we supposed to assume that the index set I of the {{mvar|A}}{{sub|k}}s is finite and has cardinality {{mvar|n}}?|date=May 2020}}. 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>\operatornamemathbb{P}</math> and {{mvar|A}} 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 {{mvar|A}}:

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

This identity is used in a simple proof of [[Markov's inequality]].

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==Mean, variance and covariance==

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> (also called "Fundamental Bridge").

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

;[[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) .</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 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}}

:"There shall correspond to each class or relation {{mvar|R}} a representing function φ({{mvar|x}}₁, ... {{mvar|x}}_n) = 0 if {{mvar|R}}({{mvar|x}}₁, ... {{mvar|x}}_n) and φ({{mvar|x}}₁, ... {{mvar|x}}_n) = 1 if ¬{{mvar|R}}({{mvar|x}}₁, ... {{mvar|x}}_n)."<ref name=Martin-1965/>{{rp|page=&nbsp;42}}(the "¬" indicates logical inversion, i.e. "NOT")

{{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>}}

[[Stephen Kleene|Kleene]] (1952)<ref>{{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> 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.

[[Stephen Kleene|Kleene]] (1952)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> 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* ... *φ_n = 0 whenever any one of the functions equals 0, it plays the role of logical OR: IF φ₁ = 0 OR φ₂ = 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 {{mvar|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. 279 ff) [[mu operator]]s (Kleene (1952)) and the CASE function (p.&nbsp;229).

For example, because the product of characteristic functions φ<sub>1</submath>\phi_1 *φ₂ \phi_2 * ...\cdots *φ_n \phi_n = 0</math> whenever any one of the functions equals {{math|0}}, it plays the role of logical OR: IF φ<sub>1</submath>\phi_1 = 0\ </math> OR φ<sub>2</submath>\ \phi_2 = 0</math> OR ... OR φ<sub>n</submath>\phi_n = 0</math> THEN their product is {{math|0}}. What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is {{math|0}} when the function {{mvar|R}} is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY,<ref (p.name=Kleene1952 />{{rp|228),}} the bounded-<ref (p.name=Kleene1952 />{{rp|228)}} and unbounded-<ref (p.name=Kleene1952 />{{rp|279 ff)}} [[mu operator]]s (Kleene (1952)) and the CASE function (p.&nbsp;<ref name=Kleene1952 />{{rp|229).}}

==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==

==Derivatives of the indicator function==

{{MainSee also|Laplacian of the indicator}}

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>

A particular indicator function is the [[Heaviside step function]]. The Heaviside step function {{mvar|H}}({{mvar|x}}) is the indicator function of the one-dimensional positive half-line, i.e. the ___domain {{closed-open|0, ∞}}. The [[distributional derivative]] of the Heaviside step function is equal to the [[Dirac delta function]], i.e.

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>

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

TheThus 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 inward [[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|''δ''<sub>''S''\delta_S(\mathbf{x})</submath>('''''x''''')}}:

with the following property:

:<math display=block>\delta_S(\mathbf{x}) = -\mathbf{n}_x \cdot \nabla_x \mathbfoperatorname\mathbb{1I}_{\!\bigl(\ \mathbf{x}\in D}\ \bigr)\ </math>

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 \mathbfoperatorname\mathbb{1I}_{\!\bigl(\ \mathbf{x}\in D}\ \bigr) \; \operatorname{d}^{n}\mathbf{x} = \oint_{S}\,f(\mathbf{\beta}) \; \operatorname{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''}}.

:<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 {{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|''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>{{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}}</ref>

:<math>-\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 ''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]] ''S''.

==See also==

{{Div col|colwidth=40em15em}}

* [[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}}

{{div col end}}

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==Notes==

jhhhht≠≥§≥≥≈″→←÷÷

{{notelist}}

==References==

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==Sources==

{{refbegin|25em}}

* {{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 |lastlast1=Cormen |firstfirst1=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}}

* {{cite book |editor-last=Davis |editor-first=Martin |editor-link=Martin Davis (mathematician) |year=1965 |title=The Undecidable |publisher=Raven Press Books |place=New York, NY}}

* {{cite book |last=Kleene |first=Stephen |author-link=Stephen Kleene |year=1971 |orig-year=1952 |title=Introduction to Metamathematics |publisher=Wolters-Noordhoff Publishing and North Holland Publishing Company |___location=Netherlands |edition=Sixth reprint, with corrections}}

* {{Cite book |lastlast1=Boolos |firstfirst1=George |title=Computability and Logic |last2=Burgess |first2=John P. |last3=Jeffrey |first3=Richard C. |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-00758-0 |___location=Cambridge UK |author-link=George Boolos |author-link2=John P. Burgess |author-link3=Richard C. Jeffrey}}

*{{cite q | Q25938993 |last1=Zadeh |first1=L.A. | author-link1 = Lotfi A. Zadeh | | journal = [[Information and Computation|Information and Control]] | doi-access = free }}

* {{cite journal |last=Zadeh |first=Lotfi A. |author-link=Lotfi A. Zadeh |date=June 1965 |title=Fuzzy sets |url=http://www-bisc.cs.berkeley.edu/zadeh/papers/Fuzzy%20Sets-1965.pdf |url-status=dead |journal=[[Information and Control]] |volume=8 |issue=3 |pages=338–353 |doi=10.1016/S0019-9958(65)90241-X |archive-url=https://web.archive.org/web/20070622151801/http://www-bisc.cs.berkeley.edu/zadeh/papers/Fuzzy%20Sets-1965.pdf |archive-date=2007-06-22|doi-access=free }}

* {{cite journal |last=Goguen |first=Joseph |author-link=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 |hdl-access=free |hdl=10338.dmlcz/103980}}

{{refend}}

[[Category:Measure theory]]