Misplaced Pages

#996003

48-403: ZCR is defined formally as where s {\displaystyle s} is a signal of length T {\displaystyle T} and 1 R < 0 {\displaystyle \mathbb {1} _{\mathbb {R} _{<0}}} is an indicator function . In some cases only the "positive-going" or "negative-going" crossings are counted, rather than all

96-436: A = b ] + [ a > b ] = 1. {\displaystyle [a<b]+[a=b]+[a>b]=1.} The Möbius function has the property (and can be defined by recurrence as ) ∑ d | n μ ( d )   =   [ n = 1 ] . {\displaystyle \sum _{d|n}\mu (d)\ =\ [n=1].} In the 1830s, Guglielmo dalla Sommaja used

144-511: A ] {\displaystyle [0\leq x\leq a]} . Following one common convention , those quantities are equal where defined: 0 0 x {\displaystyle 0^{0^{x}}} is 1 if x > 0 , is 0 if x = 0 , and is undefined otherwise. In addition to the now-standard square brackets [ · ] , and the original parentheses ( · ) , blackboard bold brackets have also been used, e.g. ⟦ · ⟧ , as well as other unusual forms of bracketing marks available in

192-457: A 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} }

240-754: A natural piecewise definition, may be expressed in terms of the Iverson bracket. The Kronecker delta notation is a specific case of Iverson notation when the condition is equality. That is, δ i j = [ i = j ] . {\displaystyle \delta _{ij}=[i=j].} The indicator function of a set A {\displaystyle A} , often denoted 1 A ( x ) {\displaystyle \mathbf {1} _{A}(x)} , I A ( x ) {\displaystyle \mathbf {I} _{A}(x)} or χ A ( x ) {\displaystyle \chi _{A}(x)} ,

288-645: 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 offers up

336-2231: A separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. We mechanically derive a well-known sum manipulation rule using Iverson brackets: ∑ k ∈ A f ( k ) + ∑ k ∈ B f ( k ) = ∑ k f ( k ) [ k ∈ A ] + ∑ k f ( k ) [ k ∈ B ] = ∑ k f ( k ) ( [ k ∈ A ] + [ k ∈ B ] ) = ∑ k f ( k ) ( [ k ∈ A ∪ B ] + [ k ∈ A ∩ B ] ) = ∑ k ∈ A ∪ B f ( k )   + ∑ k ∈ A ∩ B f ( k ) . {\displaystyle {\begin{aligned}\sum _{k\in A}f(k)+\sum _{k\in B}f(k)&;=\sum _{k}f(k)\,[k\in A]+\sum _{k}f(k)\,[k\in B]\\&;=\sum _{k}f(k)\,([k\in A]+[k\in B])\\&;=\sum _{k}f(k)\,([k\in A\cup B]+[k\in A\cap B])\\&;=\sum _{k\in A\cup B}f(k)\ +\sum _{k\in A\cap B}f(k).\end{aligned}}} The well-known rule ∑ j = 1 n ∑ k = 1 j f ( j , k ) = ∑ k = 1 n ∑ j = k n f ( j , k ) {\textstyle \sum _{j=1}^{n}\sum _{k=1}^{j}f(j,k)=\sum _{k=1}^{n}\sum _{j=k}^{n}f(j,k)}

384-1179: A similar argument, if A ≡ ∅ {\displaystyle A\equiv \emptyset } then 1 A = 0. {\displaystyle \mathbf {1} _{A}=0.} If A {\displaystyle A} and B {\displaystyle B} are two subsets of X , {\displaystyle X,} then 1 A ∩ B = min { 1 A , 1 B } = 1 A ⋅ 1 B , 1 A ∪ B = max { 1 A , 1 B } = 1 A + 1 B − 1 A ⋅ 1 B , {\displaystyle {\begin{aligned}\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},\end{aligned}}} and

432-484: Is d G ( x ) d x = − δ ( x ) {\displaystyle {\frac {dG(x)}{dx}}=-\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

480-602: Is a measurable set , then 1 A {\displaystyle \mathbf {1} _{A}} becomes a random variable whose expected value is equal to the probability of A : E ⁡ ( 1 A ) = ∫ X 1 A ( x ) d P = ∫ A d P = P ⁡ ( A ) . {\displaystyle \operatorname {E} (\mathbf {1} _{A})=\int _{X}\mathbf {1} _{A}(x)\,d\operatorname {P} =\int _{A}d\operatorname {P} =\operatorname {P} (A).} This identity

528-709: Is a stub . You can help Misplaced Pages by expanding it . Indicator function In mathematics , an indicator function or a characteristic function of a subset of a set is a function 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 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, where 1 A {\displaystyle \mathbf {1} _{A}}

SECTION 10

#1732898136997

576-648: 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}(1-\mathbf {1} _{A_{k}}(x))} is clearly a product of 0 s and 1 s. This product has the value 1 at precisely those x ∈ X {\displaystyle x\in X} that belong to none of

624-593: Is a common notation for the indicator function. Other common notations are I A , {\displaystyle I_{A},} and χ A . {\displaystyle \chi _{A}.} The indicator function of A is the Iverson bracket of the property of belonging to A ; that is, 1 A ( x ) = [ x ∈ A ] . {\displaystyle \mathbf {1} _{A}(x)=[x\in A].} For example,

672-514: Is also used to denote the characteristic function in convex analysis , which is defined as if using the reciprocal of the standard definition of the indicator function. 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 classic probability theory . For this reason, traditional probabilists use

720-1214: Is an Iverson bracket with set membership as its condition: I A ( x ) = [ x ∈ A ] . {\displaystyle \mathbf {I} _{A}(x)=[x\in A].} The Heaviside step function , sign function , and absolute value function are also easily expressed in this notation: H ( x ) = [ x ≥ 0 ] , sgn ⁡ ( x ) = [ x > 0 ] − [ x < 0 ] , {\displaystyle {\begin{aligned}H(x)&=[x\geq 0],\\\operatorname {sgn}(x)&=[x>0]-[x<0],\end{aligned}}} and | x | = x [ x > 0 ] − x [ x < 0 ] = x ( [ x > 0 ] − [ x < 0 ] ) = x ⋅ sgn ⁡ ( x ) . {\displaystyle {\begin{aligned}|x|&=x[x>0]-x[x<0]\\&=x([x>0]-[x<0])\\&=x\cdot \operatorname {sgn}(x).\end{aligned}}} The comparison functions max and min (returning

768-601: 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.} Kurt Gödel described the representing function in his 1934 paper "On undecidable propositions of formal mathematical systems" (the "¬" indicates logical inversion, i.e. "NOT"): There shall correspond to each class or relation R

816-1216: Is likewise easily derived: ∑ j = 1 n ∑ k = 1 j f ( j , k ) = ∑ j , k f ( j , k ) [ 1 ≤ j ≤ n ] [ 1 ≤ k ≤ j ] = ∑ j , k f ( j , k ) [ 1 ≤ k ≤ j ≤ n ] = ∑ j , k f ( j , k ) [ 1 ≤ k ≤ n ] [ k ≤ j ≤ n ] = ∑ k = 1 n ∑ j = k n f ( j , k ) . {\displaystyle {\begin{aligned}\sum _{j=1}^{n}\,\sum _{k=1}^{j}f(j,k)&=\sum _{j,k}f(j,k)\,[1\leq j\leq n]\,[1\leq k\leq j]\\&=\sum _{j,k}f(j,k)\,[1\leq k\leq j\leq n]\\&=\sum _{j,k}f(j,k)\,[1\leq k\leq n]\,[k\leq j\leq n]\\&=\sum _{k=1}^{n}\,\sum _{j=k}^{n}f(j,k).\end{aligned}}} For instance, Euler's totient function that counts

864-705: Is off by ⁠ 1 / 2 ⁠ for n = 1 . To get an identity valid for all positive integers n (i.e., all values for which φ ( n ) {\displaystyle \varphi (n)} is defined), a correction term involving the Iverson bracket may be added: ∑ 1 ≤ k ≤ n gcd ( k , n ) = 1 k = 1 2 n ( φ ( n ) + [ n = 1 ] ) {\displaystyle \sum _{1\leq k\leq n \atop \gcd(k,n)=1}\!\!k={\frac {1}{2}}n{\Big (}\varphi (n)+[n=1]{\Big )}} Many common functions, especially those with

912-413: 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 P {\displaystyle \operatorname {P} } and A

960-725: Is the outward normal of the surface S . This 'surface delta function' has the following property: − ∫ R n f ( x ) n x ⋅ ∇ x 1 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}\mathbf {1} _{\mathbf {x} \in D}\;d^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;d^{n-1}\mathbf {\beta } .} By setting

1008-463: Is true;}}\\0&{\text{otherwise.}}\end{cases}}} In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true. The Iverson bracket allows using capital-sigma notation without restriction on the summation index. That is, for any property P ( k ) {\displaystyle P(k)} of the integer k {\displaystyle k} , one can rewrite

SECTION 20

#1732898136997

1056-458: 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.) Given

1104-888: The Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers . The indicator function of a subset A of a set X is a function 1 A : X → { 0 , 1 } {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}} defined as 1 A ( x ) := { 1    if    x ∈ A   , 0    if    x ∉ A   . {\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1~&{\text{ if }}~x\in A~,\\0~&{\text{ if }}~x\notin A~.\end{cases}}} The Iverson bracket provides

1152-465: The free variables in that statement. This function is defined to take the value 1 for the values of the variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets: [ P ] = { 1 if  P  is true; 0 otherwise. {\displaystyle [P]={\begin{cases}1&{\text{if }}P{\text{

1200-440: The Heaviside step function is equal to the Dirac delta function , i.e. d H ( x ) d x = δ ( x ) {\displaystyle {\frac {dH(x)}{dx}}=\delta (x)} and similarly the distributional derivative of G ( x ) := 1 x < 0 {\displaystyle G(x):=\mathbf {1} _{x<0}}

1248-752: 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 ) {\displaystyle \delta _{S}(\mathbf {x} )} : δ S ( x ) = − n x ⋅ ∇ x 1 x ∈ D {\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}} where n

1296-412: The Iverson bracket is to simplify equations with special cases. For example, the formula ∑ 1 ≤ k ≤ n gcd ( k , n ) = 1 k = 1 2 n φ ( n ) {\displaystyle \sum _{1\leq k\leq n \atop \gcd(k,n)=1}\!\!k={\frac {1}{2}}n\varphi (n)} is valid for n > 1 but

1344-430: The crossings, since between a pair of adjacent positive zero-crossings there must be a single negative zero-crossing. For monophonic tonal signals, the zero-crossing rate can be used as a primitive pitch detection algorithm . Zero crossing rates are also used for Voice activity detection (VAD), which determines whether human speech is present in an audio segment or not. This signal processing -related article

1392-547: The equivalent notation, [ x ∈ A ] {\displaystyle [x\in A]} or ⟦ x ∈ A ⟧ , to be used instead of 1 A ( x ) . {\displaystyle \mathbf {1} _{A}(x)\,.} The function 1 A {\displaystyle \mathbf {1} _{A}} is sometimes denoted I A , χ A , K A , or even just A . The notation χ A {\displaystyle \chi _{A}}

1440-549: The expression 0 0 x {\displaystyle 0^{0^{x}}} to represent what now would be written [ x > 0 ] {\displaystyle [x>0]} ; he also used variants, such as ( 1 − 0 0 − x ) ( 1 − 0 0 x − a ) {\displaystyle \left(1-0^{0^{-x}}\right)\left(1-0^{0^{x-a}}\right)} for [ 0 ≤ x ≤

1488-1286: The function P ( x ) = ∏ ( 1 − f α ( x ) q − 1 ) {\textstyle 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 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 P(x)=0} . Although indicator functions are not smooth, they admit weak derivatives . For example, consider Heaviside step function H ( x ) := 1 x > 0 {\displaystyle H(x):=\mathbf {1} _{x>0}} The distributional derivative of

Zero-crossing rate - Misplaced Pages Continue

1536-464: 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 . Iverson bracket In mathematics , the Iverson bracket , named after Kenneth E. Iverson , is a notation that generalises the Kronecker delta , which is the Iverson bracket of the statement x = y . It maps any statement to a function of

1584-404: 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

1632-3550: The generalisation to arbitrary statements, notational restriction to square brackets, and applications to summation, was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions. There is a direct correspondence between arithmetic on Iverson brackets, logic, and set operations. For instance, let A and B be sets and P ( k 1 , … ) {\displaystyle P(k_{1},\dots )} any property of integers; then we have [ P ∧ Q ]   =   [ P ] [ Q ]     ; [ P ∨ Q ]   =   [ P ] + [ Q ] − [ P ] [ Q ]     ; [ ¬ P ]   =   1 − [ P ]     ; [ P  XOR  Q ]   =   | [ P ] − [ Q ] |     ; [ k ∈ A ] + [ k ∈ B ]   =   [ k ∈ A ∪ B ] + [ k ∈ A ∩ B ]     ; [ x ∈ A ∩ B ]   =   [ x ∈ A ] [ x ∈ B ]     ; [ ∀ m   : P ( k , m ) ]   =   ∏ m [ P ( k , m ) ]     ; [ ∃ m   : P ( k , m ) ]   =   min { 1 , ∑ m [ P ( k , m ) ] } = 1 − ∏ m [ ¬ P ( k , m ) ]     ; # { m | P ( k , m ) }   =   ∑ m [ P ( k , m ) ]     . {\displaystyle {\begin{aligned}[][\,P\land Q\,]~&=~[\,P\,]\,[\,Q\,]~~;\\[1em][\,P\lor Q\,]~&=~[\,P\,]\;+\;[\,Q\,]\;-\;[\,P\,]\,[\,Q\,]~~;\\[1em][\,\neg \,P\,]~&=~1-[\,P\,]~~;\\[1em][\,P{\scriptstyle {\mathsf {\text{ XOR }}}}Q\,]~&=~{\Bigl |}\,[\,P\,]\;-\;[\,Q\,]\,{\Bigr |}~~;\\[1em][\,k\in A\,]\;+\;[\,k\in B\,]~&=~[\,k\in A\cup B\,]\;+\;[\,k\in A\cap B\,]~~;\\[1em][\,x\in A\cap B\,]~&=~[\,x\in A\,]\,[\,x\in B\,]~~;\\[1em][\,\forall \,m\ :\,P(k,m)\,]~&=~\prod _{m}\,[\,P(k,m)\,]~~;\\[1em][\,\exists \,m\ :\,P(k,m)\,]~&=~\min {\Bigl \{}\;1\,,\,\sum _{m}\,[\,P(k,m)\,]\;{\Bigr \}}=1\;-\;\prod _{m}\,[\,\neg \,P(k,m)\,]~~;\\[1em]\#{\Bigl \{}\;m\,{\Big |}\,P(k,m)\;{\Bigr \}}~&=~\sum _{m}\,[\,P(k,m)\,]~~.\end{aligned}}} The notation allows moving boundary conditions of summations (or integrals) as

1680-413: The index n {\displaystyle n} of summation is understood to range over all the integers. The ramp function can be expressed R ( x ) = x ⋅ [ x ≥ 0 ] . {\displaystyle R(x)=x\cdot [x\geq 0].} The trichotomy of the reals is equivalent to the following identity: [ a < b ] + [

1728-778: The indicator function of a set is not smooth; it is continuous if and only if its support is a connected component . In the algebraic geometry of finite fields , however, every affine variety admits a ( Zariski ) continuous indicator function. Given a finite set of functions f α ∈ F q [ x 1 , … , x n ] {\displaystyle f_{\alpha }\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]} let V = { x ∈ F q n : f α ( x ) = 0 } {\displaystyle V=\left\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\right\}} be their vanishing locus. Then,

1776-482: The indicator function of the complement of A {\displaystyle A} i.e. A C {\displaystyle A^{C}} 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}}

1824-959: The larger or smaller of two arguments) may be written as max ( x , y ) = x [ x > y ] + y [ x ≤ y ] {\displaystyle \max(x,y)=x[x>y]+y[x\leq y]} and min ( x , y ) = x [ x ≤ y ] + y [ x > y ] . {\displaystyle \min(x,y)=x[x\leq y]+y[x>y].} The floor and ceiling functions can be expressed as ⌊ x ⌋ = ∑ n n ⋅ [ n ≤ x < n + 1 ] {\displaystyle \lfloor x\rfloor =\sum _{n}n\cdot [n\leq x<n+1]} and ⌈ x ⌉ = ∑ n n ⋅ [ n − 1 < x ≤ n ] , {\displaystyle \lceil x\rceil =\sum _{n}n\cdot [n-1<x\leq n],} where

1872-426: The number of positive integers up to n which are coprime to n can be expressed by φ ( n ) = ∑ i = 1 n [ gcd ( i , n ) = 1 ] , for  n ∈ N + . {\displaystyle \varphi (n)=\sum _{i=1}^{n}[\gcd(i,n)=1],\qquad {\text{for }}n\in \mathbb {N} ^{+}.} Another use of

1920-532: The predicate is replaced by a quantity interpreted as the degree of truth. The indicator or characteristic function of a subset A of some set X maps elements of X to the range { 0 , 1 } {\displaystyle \{0,1\}} . This mapping is surjective only when A is a non-empty proper subset of X . If A ≡ X , {\displaystyle A\equiv X,} then 1 A = 1. {\displaystyle \mathbf {1} _{A}=1.} By

1968-900: 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|}

Zero-crossing rate - Misplaced Pages Continue

2016-434: 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. In general,

2064-586: 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, the bounded- and unbounded- mu operators and the CASE function. 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

2112-446: The restricted sum ∑ k : P ( k ) f ( k ) {\displaystyle \sum _{k:P(k)}f(k)} in the unrestricted form ∑ k f ( k ) ⋅ [ P ( k ) ] {\displaystyle \sum _{k}f(k)\cdot [P(k)]} . With this convention, f ( k ) {\displaystyle f(k)} does not need to be defined for

2160-493: 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 ϕ 1 ∗ ϕ 2 ∗ ⋯ ∗ ϕ n = 0 {\displaystyle \phi _{1}*\phi _{2}*\cdots *\phi _{n}=0} whenever any one of

2208-543: 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

2256-410: 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 and modern many-valued logic , predicates are the characteristic functions of a probability distribution . That is, the strict true/false valuation of

2304-487: The values of k for which the Iverson bracket equals 0 ; that is, a summand f ( k ) [ false ] {\displaystyle f(k)[{\textbf {false}}]} must evaluate to 0 regardless of whether f ( k ) {\displaystyle f(k)} is defined. The notation was originally introduced by Kenneth E. Iverson in his programming language APL , though restricted to single relational operators enclosed in parentheses, while

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