Characteristic function: Difference between revisions

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SomeIn mathematicians[[set usetheory]] theand phrasein '''''characteristicmost function'''''areas synonymouslyof withMathematics the ''[[indicator'characteristic function]]''. The indicator function' of a [[subset]] ''A'' of a [[set]] ''BX'' is the [[function (mathematics)|function]] <math>\chi_A: X \to \{0, 1\}</math> with ___domain ''BX'', whosehaving value is 1 at eachpoints point inof ''A'' and 0 at eachpoints point that is inof ''B''X but- not in ''A''.

:<math>\chi_A(x) = \begin{cases}1, &x \in A,\\0, &x \in X-A\end{cases}</math>

Sometimes this is called '''indicator function''', or simply '''indicator'''. An example of characteristic function is the [[Heaviside function]].

Elementary properties of the characteristic function are the following:

<math>\chi_{\emptyset} \equiv 0\ \ \ \ \ \chi_X \equiv 1\ \ \ \ \ \chi_{A\cap B}= \chi_A \chi_B \ \ \ \ \ \chi_{A\cup B}= \chi_A + \chi_B - \chi_A \chi_B</math>

<math>\chi_{X -A}= 1 -\chi_A \ \ \ \ \ \chi_{A - B}= \chi_A - \chi_A \chi_B \ \ \ \ \ \chi_{A \Delta B}= \chi_A + \chi_B - 2 \chi_A \chi_B</math>

where <math>\emptyset</math> is the empty set and <math>A \Delta B = A\cup B \ - \ A\cap B</math> is the [[symmetric difference]] of <math>A</math> and <math>B</math>.

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