Indicator Function

The indicator function of a subset

A

of a set

X

is a function

:

\mathbf{1}_A : X 	o \lbrace 0,1 
brace \,

defined as

:

\mathbf{1}_A(x) = 
\left\{\begin{matrix} 
1 &\mbox{if}\ x \in A, \
0 &\mbox{if}\ x 
otin A.
\end{matrix}
ight.

The Iverson Bracket allows the notation

\in A

.

The indicator function of

A

is sometimes denoted
:

\chi_A(x)

\mathbf{I}_A(x)

or even

A(x).

(The Greek Letter χ because it is the initial letter of the Greek Etymon of the word ''characteristic''.)

WARNINGS

The notation $\mathbf{1}_A$ may signify the Identity Function .

The notation $\chi_{A}$ may signify the Characteristic Function in Convex Analysis .

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

Boolos, Burgess, and Jeffrey (2002) define the ''characteristic function'' as follows:
:"The ''characteristic function'' of a k-place relation is the k-argument function that takes the value 1 for a k-tuple if the relation holds of the k-tuple, and the value 0 if it does not; and a relation is ''effectively decidable'' if its characteristic function is effectively computable, and is ''(primitive) recursive'' if its characteristic function is (primitive) recursive." (italics in original, p.73–74)

The mapping which associates a subset

A

X

to its ''indicator function''

\mathbf{1}_A

is Injective ; its range is the set of functions

f : X 	o \{0,1\}

1 = 1, 1---0 = 0 etc, and likewise the "+" and "-" represent algebraic addition and subtraction. 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,\,

	CATEGORIES ABOUT INDICATOR FUNCTION

	measure theory

	integral calculus

	real analysis

	discrete mathematics

	mathematical logic

	basic concepts in set theory

	probability theory

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Information About

Indicator Function