Characteristic function: Difference between revisions
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In [[set theory]], the '''characteristic function''' or '''indicator function''' of a [[subset]] ''A'' of a [[set (mathematics)|set]] ''X'' is the function, often denoted χ<sub>''A''</sub> or ''I''<sub>''A''</sub>, from ''X'' to the set {0,1} which takes the value 1 on elements of ''A'' and 0 otherwise. | In [[set theory]], the '''characteristic function''' or '''indicator function''' of a [[subset]] ''A'' of a [[set (mathematics)|set]] ''X'' is the function, often denoted χ<sub>''A''</sub> or ''I''<sub>''A''</sub>, from ''X'' to the set {0,1} which takes the value 1 on elements of ''A'' and 0 otherwise. | ||
We can express elementary set-theoretic operations in terms of characteristic functions: | |||
*[[Empty set]]: <math>\chi_\emptyset = 0 ;\,</math> | |||
*[[Intersection]]: <math>\chi_{A \cap B} = \min\{\chi_A,\chi_B\} = \chi_A \cdot \chi_B ;\,</math> | |||
*[[Union]]: <math>\chi_{A \cup B} = \max\{\chi_A,\chi_B\} = \chi_A + \chi_B - \chi_A \cdot \chi_B ;\,</math> | |||
*[[Symmetric difference]]: <math>\chi_{A \bigtriangleup B} = \chi_A + \chi_B \pmod 2 .\,</math> | |||
Revision as of 14:15, 7 December 2008
In set theory, the characteristic function or indicator function of a subset A of a set X is the function, often denoted χA or IA, from X to the set {0,1} which takes the value 1 on elements of A and 0 otherwise.
We can express elementary set-theoretic operations in terms of characteristic functions:
