# Indicator function

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Mathematical function characterizing set membership

This article is about the 0–1 indicator function. For the 0–infinity indicator function, see [characteristic function (convex analysis)](/source/Characteristic_function_(convex_analysis)).

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A three-dimensional plot of an indicator function, shown over a square two-dimensional domain (set X): the "raised" portion overlays those two-dimensional points which are members of the "indicated" subset (A).

In [mathematics](/source/Mathematics), an **indicator function** or a **characteristic function** of a [subset](/source/Subset) of a [set](/source/Set_(mathematics)) is a [function](/source/Function_(mathematics)) that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then the indicator function of A is the function 1 A {\displaystyle \mathbf {1} _{A}} defined by 1 A ( x ) = 1 {\displaystyle \mathbf {1} _{A}\!(x)=1} if x ∈ A , {\displaystyle x\in A,} and 1 A ( x ) = 0 {\displaystyle \mathbf {1} _{A}\!(x)=0} otherwise. Other common notations are 𝟙*A* and χ A . {\displaystyle \chi _{A}.} [a]

The indicator function of A is the [Iverson bracket](/source/Iverson_bracket) of the property of belonging to A; that is,

1 A ( x ) = [ x ∈ A ] . {\displaystyle \mathbf {1} _{A}(x)=\left[\ x\in A\ \right].}

For example, the [Dirichlet function](/source/Dirichlet_function) is the indicator function of the [rational numbers](/source/Rational_number) as a subset of the [real numbers](/source/Real_number).

## Definition

Given an arbitrary set X, the indicator function of a subset A of X is the function 1 A : X → { 0 , 1 } {\displaystyle \mathbf {1} _{A}\colon X\rightarrow \{0,1\}} defined by [1 A ( x ) = { 1 if x ∈ A 0 if x ∉ A . {\displaystyle \operatorname {\mathbf {1} } _{A}\!(x)={\begin{cases}1&{\text{if }}x\in A\\0&{\text{if }}x\notin A\,.\end{cases}}}](https://en.wikipedia.org/w/index.php?title=Special:MathWikibase&qid=Q371983)

The [Iverson bracket](/source/Iverson_bracket) provides the equivalent notation [ x ∈ A ] {\displaystyle \left[\ x\in A\ \right]} or ⟦ *x* ∈ *A* ⟧, that can be used instead of 1 A ( x ) . {\displaystyle \mathbf {1} _{A}\!(x).}

The function 1 A {\displaystyle \mathbf {1} _{A}} is sometimes denoted 𝟙*A*, IA, χA[a] or even just A.[b]

## Notation and terminology

The notation χ A {\displaystyle \chi _{A}} is also used to denote the [characteristic function](/source/Characteristic_function_(convex_analysis)) in [convex analysis](/source/Convex_analysis), which is defined as if using the [reciprocal](/source/Multiplicative_inverse) of the standard definition of the indicator function.

A related concept in [statistics](/source/Statistics) is that of a [dummy variable](/source/Dummy_variable_(statistics)). (This must not be confused with "dummy variables" as that term is usually used in mathematics, also called a [bound variable](/source/Free_variables_and_bound_variables).)

The term "[characteristic function](/source/Characteristic_function_(probability_theory))" has an unrelated meaning in [classic probability theory](/source/Probability_theory). For this reason, [traditional probabilists](/source/List_of_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](/source/Fuzzy_logic) and [modern many-valued logic](/source/Many-valued_logic), predicates are the [characteristic functions](/source/Characteristic_function_(probability_theory)) of a [probability distribution](/source/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](/source/Function_(mathematics)) of a subset A of some set X [maps](/source/Map_(mathematics)) elements of X to the [codomain](/source/Codomain) { 0 , 1 } . {\displaystyle \{0,\,1\}.}

This mapping is [surjective](/source/Surjective) only when A is a non-empty [proper subset](/source/Proper_subset) of X. If A = X , {\displaystyle A=X,} then 1 A ≡ 1. {\displaystyle \mathbf {1} _{A}\equiv 1.} By a similar argument, if A = ∅ {\displaystyle A=\emptyset } then 1 A ≡ 0. {\displaystyle \mathbf {1} _{A}\equiv 0.}

If A {\displaystyle A} and B {\displaystyle B} are two subsets of X , {\displaystyle X,} then 1 A ∩ B ( x ) = min { 1 A ( x ) , 1 B ( x ) } = 1 A ( x ) ⋅ 1 B ( x ) , 1 A ∪ B ( x ) = max { 1 A ( x ) , 1 B ( x ) } = 1 A ( x ) + 1 B ( x ) − 1 A ( x ) ⋅ 1 B ( x ) , {\displaystyle {\begin{aligned}\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 {\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{aligned}}}

and the indicator function of the [complement](/source/Complement_(set_theory)) of A {\displaystyle A} i.e. A ∁ {\displaystyle A^{\complement }} is: 1 A ∁ = 1 − 1 A . {\displaystyle \mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}.}

More generally, suppose A 1 , … , A n {\displaystyle A_{1},\dotsc ,A_{n}} is a collection of subsets of X. For any x ∈ X : {\displaystyle x\in X:}

∏ k ∈ I ( 1 − 1 A k ( x ) ) {\displaystyle \prod _{k\in I}\left(\ 1-\mathbf {1} _{A_{k}}\!\left(x\right)\ \right)}

is a product of 0s and 1s. This product has the value 1 at precisely those x ∈ X {\displaystyle x\in X} that belong to none of the sets A k {\displaystyle A_{k}} and is 0 otherwise. That is

∏ k ∈ I ( 1 − 1 A k ) = 1 X − ⋃ k A k = 1 − 1 ⋃ k A k . {\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.}

Expanding the product on the left hand side,

1 ⋃ k A k = 1 − ∑ F ⊆ { 1 , 2 , … , n } ( − 1 ) | F | 1 ⋂ F A k = ∑ ∅ ≠ F ⊆ { 1 , 2 , … , n } ( − 1 ) | F | + 1 1 ⋂ F A k {\displaystyle \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}}}

where | F | {\displaystyle |F|} is the [cardinality](/source/Cardinality) of F. This is one form of the principle of [inclusion-exclusion](/source/Inclusion-exclusion).

As suggested by the previous example, the indicator function is a useful notational device in [combinatorics](/source/Combinatorics). The notation is used in other places as well, for instance in [probability theory](/source/Probability_theory): if X is a [probability space](/source/Probability_space) with probability measure P {\displaystyle \mathbb {P} } and A is a [measurable set](/source/Measure_(mathematics)), then 1 A {\displaystyle \mathbf {1} _{A}} becomes a [random variable](/source/Random_variable) whose [expected value](/source/Expected_value) is equal to the probability of A:

E X ⁡ { 1 A ( x ) } = ∫ X 1 A ( x ) d P ⁡ ( x ) = ∫ A d P ⁡ ( x ) = P ⁡ ( A ) . {\displaystyle \operatorname {\mathbb {E} } _{X}\left\{\ \mathbf {1} _{A}(x)\ \right\}\ =\ \int _{X}\mathbf {1} _{A}(x)\ \operatorname {d\ \mathbb {P} } (x)=\int _{A}\operatorname {d\ \mathbb {P} } (x)=\operatorname {\mathbb {P} } (A).}

This identity is used in a simple proof of [Markov's inequality](/source/Markov's_inequality).

In many cases, such as [order theory](/source/Order_theory), the inverse of the indicator function may be defined. This is commonly called the [generalized Möbius function](/source/Generalized_M%C3%B6bius_function), as a generalization of the inverse of the indicator function in elementary [number theory](/source/Number_theory), the [Möbius function](/source/M%C3%B6bius_function). (See paragraph below about the use of the inverse in classical recursion theory.)

## Mean, variance and covariance

Given a [probability space](/source/Probability_space) ( Ω , F , P ) {\displaystyle \textstyle (\Omega ,{\mathcal {F}},\operatorname {P} )} with A ∈ F , {\displaystyle A\in {\mathcal {F}},} the indicator random variable 1 A : Ω → R {\displaystyle \mathbf {1} _{A}\colon \Omega \rightarrow \mathbb {R} } is defined by 1 A ( ω ) = 1 {\displaystyle \mathbf {1} _{A}(\omega )=1} if ω ∈ A , {\displaystyle \omega \in A,} otherwise 1 A ( ω ) = 0. {\displaystyle \mathbf {1} _{A}(\omega )=0.}

**[Mean](/source/Mean)**
- E ⁡ ( 1 A ( ω ) ) = P ⁡ ( A ) {\displaystyle \ \operatorname {\mathbb {E} } (\mathbf {1} _{A}(\omega ))=\operatorname {\mathbb {P} } (A)\ } (also called "Fundamental Bridge").

**[Variance](/source/Variance)**
- Var ⁡ ( 1 A ( ω ) ) = P ⁡ ( A ) ( 1 − P ⁡ ( A ) ) . {\displaystyle \ \operatorname {Var} (\mathbf {1} _{A}(\omega ))=\operatorname {\mathbb {P} } (A)(1-\operatorname {\mathbb {P} } (A)).}

**[Covariance](/source/Covariance)**
- Cov ⁡ ( 1 A ( ω ) , 1 B ( ω ) ) = P ⁡ ( A ∩ B ) − P ⁡ ( A ) P ⁡ ( B ) . {\displaystyle \ \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](/source/Kurt_G%C3%B6del) described the *representing function* in his 1934 paper "On undecidable propositions of formal mathematical systems" (the symbol "¬" indicates logical inversion, i.e. "NOT"):[1]: 42

There shall correspond to each class or relation R a representing function ϕ ( x 1 , … x n ) = 0 {\displaystyle \phi (x_{1},\ldots x_{n})=0} if R ( x 1 , … x n ) {\displaystyle R(x_{1},\ldots x_{n})} and ϕ ( x 1 , … x n ) = 1 {\displaystyle \phi (x_{1},\ldots x_{n})=1} if ¬ R ( x 1 , … x n ) . {\displaystyle \neg R(x_{1},\ldots x_{n}).}

[Kleene](/source/Stephen_Kleene) offers up the same definition in the context of the [primitive recursive functions](/source/Primitive_recursive_function) as a function φ of a predicate P takes on values 0 if the predicate is true and 1 if the predicate is false.[2]

For example, because the product of characteristic functions ϕ 1 ∗ ϕ 2 ∗ ⋯ ∗ ϕ n = 0 {\displaystyle \phi _{1}*\phi _{2}*\cdots *\phi _{n}=0} whenever any one of the functions equals 0, it plays the role of logical OR: IF ϕ 1 = 0 {\displaystyle \phi _{1}=0\ } OR ϕ 2 = 0 {\displaystyle \ \phi _{2}=0} OR ... OR ϕ n = 0 {\displaystyle \phi _{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,[2]: 228 the bounded-[2]: 228 and unbounded-[2]: 279 ff [mu operators](/source/Mu_operator) and the CASE function.[2]: 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](/source/Fuzzy_set_theory)*, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in some [algebra](/source/Universal_algebra) or [structure](/source/Structure_(mathematical_logic)) (usually required to be at least a [poset](/source/Partially_ordered_set) or [lattice](/source/Lattice_(order))). Such generalized characteristic functions are more usually called [membership functions](/source/Membership_function_(mathematics)), and the corresponding "sets" are called *fuzzy* sets. Fuzzy sets model the gradual change in the membership [degree](/source/Degree_of_truth) seen in many real-world [predicates](/source/Predicate_(mathematics)) like "tall", "warm", etc.

## Smoothness

See also: [Laplacian of the indicator](/source/Laplacian_of_the_indicator)

In general, the indicator function of a set is not smooth; it is continuous if and only if its [support](/source/Support_(math)) is a [connected component](/source/Connected_component_(topology)). In the [algebraic geometry](/source/Algebraic_geometry) of [finite fields](/source/Finite_fields), however, every [affine variety](/source/Affine_variety) admits a ([Zariski](/source/Zariski_topology)) continuous indicator function.[3] Given a [finite set](/source/Finite_set) of functions f α ∈ F q [ x 1 , … , x n ] {\displaystyle f_{\alpha }\in \mathbb {F} _{q}\left[\ x_{1},\ldots ,x_{n}\right]} let V = { x ∈ F q n : f α ( x ) = 0 } {\displaystyle V={\bigl \{}\ x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\ {\bigr \}}} be their vanishing locus. Then, the function P ( x ) = ∏ ( 1 − f α ( x ) q − 1 ) {\textstyle \mathbb {P} (x)=\prod \left(\ 1-f_{\alpha }(x)^{q-1}\right)} acts as an indicator function for V . {\displaystyle V.} If x ∈ V {\displaystyle x\in V} then P ( x ) = 1 , {\displaystyle \mathbb {P} (x)=1,} otherwise, for some f α , {\displaystyle f_{\alpha },} we have f α ( x ) ≠ 0 {\displaystyle f_{\alpha }(x)\neq 0} which implies that f α ( x ) q − 1 = 1 , {\displaystyle f_{\alpha }(x)^{q-1}=1,} hence P ( x ) = 0. {\displaystyle \mathbb {P} (x)=0.}

Although indicator functions are not smooth, they admit [weak derivatives](/source/Weak_derivative). For example, consider [Heaviside step function](/source/Heaviside_step_function) H ( x ) ≡ I ( x > 0 ) {\displaystyle H(x)\equiv \operatorname {\mathbb {I} } \!{\bigl (}x>0{\bigr )}} The [distributional derivative](/source/Distributional_derivative) of the Heaviside step function is equal to the [Dirac delta function](/source/Dirac_delta_function), i.e. d H ( x ) d x = δ ( x ) {\displaystyle {\frac {\mathrm {d} H(x)}{\mathrm {d} x}}=\delta (x)} and similarly the distributional derivative of G ( x ) := I ( x < 0 ) {\displaystyle G(x):=\operatorname {\mathbb {I} } \!{\bigl (}x<0{\bigr )}} is d G ( x ) d x = − δ ( x ) . {\displaystyle {\frac {\mathrm {d} G(x)}{\mathrm {d} x}}=-\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 D. The surface of D will be denoted by S. Proceeding, it can be derived that the inward [normal derivative](/source/Normal_derivative) of the indicator gives rise to a *[surface delta function](/source/Surface_delta_function)*, which can be indicated by δ S ( x ) {\displaystyle \delta _{S}(\mathbf {x} )} : δ S ( x ) = − n x ⋅ ∇ x I ( x ∈ D ) {\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\operatorname {\mathbb {I} } \!{\bigl (}\ \mathbf {x} \in D\ {\bigr )}\ } where n is the outward [normal](/source/Normal_(geometry)) of the surface S. This 'surface delta function' has the following property:[4] − ∫ R n f ( x ) n x ⋅ ∇ x I ( x ∈ D ) d n ⁡ x = ∮ S f ( β ) d n − 1 ⁡ β . {\displaystyle -\int _{\mathbb {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\operatorname {\mathbb {I} } \!{\bigl (}\ \mathbf {x} \in D\ {\bigr )}\;\operatorname {d} ^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;\operatorname {d} ^{n-1}\mathbf {\beta } .}

By setting the function f equal to one, it follows that the [inward normal derivative of the indicator](/source/Laplacian_of_the_indicator#Dirac_surface_delta_function) integrates to the numerical value of the [surface area](/source/Surface_area) S.

## See also

- [Dirac measure](/source/Dirac_measure)

- [Laplacian of the indicator](/source/Laplacian_of_the_indicator)

- [Dirac delta](/source/Dirac_delta)

- [Extension (predicate logic)](/source/Extension_(predicate_logic))

- [Free variables and bound variables](/source/Free_variables_and_bound_variables)

- [Heaviside step function](/source/Heaviside_step_function)

- [Identity function](/source/Identity_function)

- [Iverson bracket](/source/Iverson_bracket)

- [Kronecker delta](/source/Kronecker_delta), a function that can be viewed as an indicator for the [identity relation](/source/Equality_(mathematics))

- [Macaulay brackets](/source/Macaulay_brackets)

- [Multiset](/source/Multiset)

- [Membership function](/source/Membership_function_(mathematics))

- [Simple function](/source/Simple_function)

- [Dummy variable (statistics)](/source/Dummy_variable_(statistics))

- [Statistical classification](/source/Statistical_classification)

- [Zero-one loss function](/source/Zero-one_loss_function)

- [Subobject classifier](/source/Subobject_classifier), a related concept from [topos theory](/source/Topos_theory).

## Notes

1. ^ [***a***](#cite_ref-χαρακτήρ_1-0) [***b***](#cite_ref-χαρακτήρ_1-1) The [Greek letter](/source/Greek_alphabet) χ appears because it is the initial letter of the Greek word χαρακτήρ, which is the ultimate origin of the word *characteristic*.

1. **[^](#cite_ref-2)** The set of all indicator functions on X can be identified with the set operator P ( X ) , {\displaystyle {\mathcal {P}}(X),} the [power set](/source/Power_set) of X. Consequently, both sets are denoted by the conventional [abuse of notation](/source/Abuse_of_notation) as 2 X , {\displaystyle 2^{X},} in analogy to the relation for the count of elements in the powerset and the original set. This is a special case ( Y = { 0 , 1 } ) {\displaystyle \left(Y=\{0,\,1\}\right)} of the notation Y X {\displaystyle Y^{X}} for the set of all functions f {\displaystyle f} such that f : X ↦ Y . {\displaystyle f:X\mapsto Y\,.}

## References

1. **[^](#cite_ref-Martin-1965_3-0)** [Davis, Martin](/source/Martin_Davis_(mathematician)), ed. (1965). *The Undecidable*. New York, NY: Raven Press Books. pp. 41–74.

1. ^ [***a***](#cite_ref-Kleene1952_4-0) [***b***](#cite_ref-Kleene1952_4-1) [***c***](#cite_ref-Kleene1952_4-2) [***d***](#cite_ref-Kleene1952_4-3) [***e***](#cite_ref-Kleene1952_4-4) [Kleene, Stephen](/source/Stephen_Kleene) (1971) [1952]. *Introduction to Metamathematics* (Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company. p. 227.

1. **[^](#cite_ref-5)** Serre. *Course in Arithmetic*. p. 5.

1. **[^](#cite_ref-6)** Lange, Rutger-Jan (2012). "Potential theory, path integrals and the Laplacian of the indicator". *Journal of High Energy Physics*. **2012** (11): 29–30. [arXiv](/source/ArXiv_(identifier)):[1302.0864](https://arxiv.org/abs/1302.0864). [Bibcode](/source/Bibcode_(identifier)):[2012JHEP...11..032L](https://ui.adsabs.harvard.edu/abs/2012JHEP...11..032L). [doi](/source/Doi_(identifier)):[10.1007/JHEP11(2012)032](https://doi.org/10.1007%2FJHEP11%282012%29032). [S2CID](/source/S2CID_(identifier)) [56188533](https://api.semanticscholar.org/CorpusID:56188533).

## Sources

- Folland, G.B. (1999). *Real Analysis: Modern Techniques and Their Applications* (Second ed.). John Wiley & Sons, Inc. [ISBN](/source/ISBN_(identifier)) [978-0-471-31716-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-31716-6).

- [Cormen, Thomas H.](/source/Thomas_H._Cormen); [Leiserson, Charles E.](/source/Charles_E._Leiserson); [Rivest, Ronald L.](/source/Ronald_L._Rivest); [Stein, Clifford](/source/Clifford_Stein) (2001). "Section 5.2: Indicator random variables". [*Introduction to Algorithms*](/source/Introduction_to_Algorithms) (Second ed.). MIT Press and McGraw-Hill. pp. [94](https://archive.org/details/introductiontoal00corm_691/page/n116)–99. [ISBN](/source/ISBN_(identifier)) [978-0-262-03293-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-03293-3).

- [Davis, Martin](/source/Martin_Davis_(mathematician)), ed. (1965). *The Undecidable*. New York, NY: Raven Press Books.

- [Kleene, Stephen](/source/Stephen_Kleene) (1971) [1952]. *Introduction to Metamathematics* (Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company.

- [Boolos, George](/source/George_Boolos); [Burgess, John P.](/source/John_P._Burgess); [Jeffrey, Richard C.](/source/Richard_C._Jeffrey) (2002). *Computability and Logic*. Cambridge UK: Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-0-521-00758-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-00758-0).

- [Zadeh, L.A.](/source/Lotfi_A._Zadeh) (June 1965). ["Fuzzy sets"](https://doi.org/10.1016%2FS0019-9958%2865%2990241-X). *[Information and Control](/source/Information_and_Computation)*. **8** (3). San Diego: 338–353. [doi](/source/Doi_(identifier)):[10.1016/S0019-9958(65)90241-X](https://doi.org/10.1016%2FS0019-9958%2865%2990241-X). [ISSN](/source/ISSN_(identifier)) [0019-9958](https://search.worldcat.org/issn/0019-9958). [Zbl](/source/Zbl_(identifier)) [0139.24606](https://zbmath.org/?format=complete&q=an:0139.24606). [Wikidata](/source/WDQ_(identifier)) [Q25938993](https://www.wikidata.org/wiki/Q25938993).

- [Goguen, Joseph](/source/Joseph_Goguen) (1967). "*L*-fuzzy sets". *Journal of Mathematical Analysis and Applications*. **18** (1): 145–174. [doi](/source/Doi_(identifier)):[10.1016/0022-247X(67)90189-8](https://doi.org/10.1016%2F0022-247X%2867%2990189-8). [hdl](/source/Hdl_(identifier)):[10338.dmlcz/103980](https://hdl.handle.net/10338.dmlcz%2F103980).

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Adapted from the Wikipedia article [Indicator function](https://en.wikipedia.org/wiki/Indicator_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Indicator_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.