Characteristic function (mathematics)

In Mathematical, a characteristic function , or indicating function , is a function definite on a Ensemble E which clarifies the membership or not of a subset F of E of any element of E .

Formally, the function characteristic of a subset F of a unit E is a function:

\begin{array}{rcl} \chi_F : E & \ longrightarrow & \ {0,1 \} \ \

X & \ longmapsto & \ left \ {\ begin {matrix} 1 \ \ mbox {if} \ X \ \ in \ F \ \ 0 \ \ mbox {if} \ X \ \ notin \ F \ end {matrix} \ right. \end{array}

The function characteristic of F is often noted $\ chi_F$ or 1_F.

For example, the function of Dirichlet is the function characteristic of $\ mathbb {Q}$ in $\ mathbb {R}$ : it is defined on $\ mathbb {R}$ and is worth 1 if X is rational, 0 if not. As $\ mathbb {Q}$ is dense in $\ mathbb {R}$ , it is a function everywhere discontinuous.

Caution

Under the probable influence of English ( indicator function ) the term of indicating function is sometimes used for characteristic function. This denomination also has the advantage of avoiding confusion with the characteristic function used of probability.

The function 1_F can indicate the Fonction identity.

Properties

If has and B is two subsets of E then

\ chi_ {has \ course B} = \ min \ {\ chi_A, \ chi_B \} = \ chi_A \ times \ chi_B

\ chi_ {has \ cup B} = \ max \ = \ chi_A + \ chi_B - \ chi_A \ times \ chi_B

\ chi_ {has \ triangle B} = \ chi_A + \ chi_B - 2 \ chi_A \ times \ chi_B

A \ subseteq B \ Leftrightarrow \ chi_ {has} \ the \ chi_ {B}

See too

Analysis
Measurement

References

Folland, G.B. ; Real Analysis: Modern Techniques and Their Applications, 2nd ED, John Wiley & Sounds, Inc., 1999.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms , Second Edition. MIT Close and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 5.2: Indicator random variable, pp.94– 99.
Martin Davis ED. (1965), The Undecidable , Raven Close Books, Ltd., New York.
Stephen Kleene, (1952), Introduction to Metamathematics , Wolters-Noordhoff Publishing and North Holland Publishing Company, Netherlands, Sixth Reprint with corrections 1971.
George Boolos, John P. Burgess, Richard C. Jeffrey (2002), Cambridge University Close, Cambridge the U.K., ISBN 0-521-00758-5.
Lotfi A. Zadeh, 1965, " Fuzzy sets". Information and Control 8 : 338-353. * Joseph Goguen, 1967, " L - fuzzy sets". Newspaper off Mathematical Analysis and Applications 18 : 145-174

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