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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

$\mathbf {1} _{A}\colon X\to \{0,1\}\,$

defined as

$\mathbf {1} _{A}(x):={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x\notin A.\end{cases}}$

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 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.Template:Clarify
• The notation χ_A is also used to denote the characteristic function in convex analysis.Template:Clarify

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 X to the 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. "$\cap$" and "$\cup$" is intersection and union, respectively.

If $A$ and $B$ are two subsets of $X$, then

$\mathbf {1} _{A\cap B}=\min\{\mathbf {1} _{A},\mathbf {1} _{B}\}=\mathbf {1} _{A}\cdot \mathbf {1} _{B},$

$\mathbf {1} _{A\cup B}=\max\{{\mathbf {1} _{A},\mathbf {1} _{B}}\}=\mathbf {1} _{A}+\mathbf {1} _{B}-\mathbf {1} _{A}\cdot \mathbf {1} _{B},$

and the indicator function of the complement of $A$ i.e. $A^{C}$ is:

$\mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}$.

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

$\prod _{k\in I}(1-\mathbf {1} _{A_{k}}(x))$

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 A_k and is 0 otherwise. That is

$\prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.$

Expanding the product on the left hand side,

$\mathbf {1} _{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|}\mathbf {1} _{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|+1}\mathbf {1} _{\bigcap _{F}A_{k}}$

where |F| 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 $\mathbb {P}$ and $A$ is a measurable set, then $\mathbf {1} _{A}$ becomes a random variable whose expected value is equal to the probability of $A$:

$\operatorname {E} (\mathbf {1} _{A})=\int _{X}\mathbf {1} _{A}(x)\,d\mathbb {P} =\int _{A}d\mathbb {P} =\operatorname {P} (A)$.

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 $\textstyle (\Omega ,{\mathcal {F}},\mathbb {P} )$ with $A\in {\mathcal {F}}$, the indicator random variable $\mathbf {1} _{A}\colon \Omega \rightarrow {\mathbb {R}}$ is defined by $\mathbf {1} _{A}(\omega )=1$ if $\omega \in A,$ otherwise $\mathbf {1} _{A}(\omega )=0.$

Mean

$\operatorname {E} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)$

Variance

$\operatorname {Var} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)(1-\operatorname {P} (A))$

Covariance

$\operatorname {Cov} (\mathbf {1} _{A}(\omega ),\mathbf {1} _{B}(\omega ))=\operatorname {P} (A\cap B)-\operatorname {P} (A)\operatorname {P} (B)$

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 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, . . ., x_n) = 0 if R(x₁, . . ., x_n) and φ(x₁, . . ., x_n) = 1 if ~R(x₁, . . ., x_n)." (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 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 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.

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.

$\delta (x)={\tfrac {dH(x)}{dx}},$

with the following property:

$\int _{-\infty }^{\infty }f(x)\,\delta (x)dx=f(0).$

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 of the indicator gives rise to a 'surface delta function', which can be indicated by δ_S(x):

$\delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}$

where n is the outward normal of the surface S. This 'surface delta function' has the following property:

$-\int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}\;d^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;d^{n-1}\mathbf {\beta } .$

By setting the function f equal to one, it follows that the 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 identity relation
• Multiset
• Membership function
• Simple function
• Dummy variable (statistics)

Notes

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References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang