PlanetMath: characteristic function

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characteristic function

(Definition)

Definition Suppose is a subset of a set . Then the function

$\displaystyle \chi_A(x) = \begin{cases}1,&\text{when }x\in A,\\ 0,&\text{when }x\in X\setminus A \end{cases}$

is the characteristic function for

Suppose

are subsets of a set

For set intersections and set unions, we have

$\displaystyle \chi_{A\cap B}$ $\displaystyle =$ $\displaystyle \chi_A \chi_B,$

$\displaystyle \chi_{A\cup B}$ $\displaystyle =$ $\displaystyle \chi_A + \chi_B - \chi_{A\cap B},$

$\displaystyle \chi_{A\cap B}$ $\displaystyle =$ $\displaystyle \min(\chi_A,\chi_B),$

$\displaystyle \chi_{A\cup B}$ $\displaystyle =$ $\displaystyle \max(\chi_A,\chi_B).$
For the symmetric difference,
$\displaystyle \chi_{A\bigtriangleup B} = \chi_A + \chi_B - 2\chi_{A\cap B}.$
For the set complement,
$\displaystyle \chi_{A^\complement} = 1-\chi_A.$

A synonym for characteristic function is indicator function [1].

Bibliography

1: G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

"characteristic function" is owned by bbukh. [ full author list (3) | owner history (2) ]

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Other names:

indicator function

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Cross-references: complement, symmetric difference, unions, intersections, function, subset
There are 40 references to this entry.

This is version 7 of characteristic function, born on 2001-10-18, modified 2004-11-04.
Object id is 350, canonical name is CharacteristicFunction.
Accessed 17437 times total.

Classification:

AMS MSC:	03-00 (Mathematical logic and foundations :: General reference works )
	26-00 (Real functions :: General reference works )
	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	28-00 (Measure and integration :: General reference works )

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