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characteristic function
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(Definition)
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Definition Suppose $A$ is a subset of a set $X$ . Then the function
is the characteristic function for $A$ .
Suppose $A,B$ are subsets of a set $X$ .
- For set intersections and set unions, we have \begin{eqnarray*} \chi_{A\cap B} &=& \chi_A \chi_B, \\ \chi_{A\cup B} &=& \chi_A + \chi_B - \chi_{A\cap B},\\ \chi_{A\cap B} &=& \min(\chi_A,\chi_B),\\ \chi_{A\cup B} &=& \max(\chi_A,\chi_B). \end{eqnarray*}
- For the symmetric difference, $$\chi_{A\bigtriangleup B} = \chi_A + \chi_B - 2\chi_{A\cap B}.$$
- For the set complement, $$\chi_{A^\complement} = 1-\chi_A. $$
A synonym for characteristic function is indicator function [1].
- 1
- G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
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"characteristic function" is owned by bbukh. [ full author list (3) | owner history (2) ]
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Cross-references: complement, symmetric difference, unions, intersections, function, subset
There are 32 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 21486 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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