Characteristic function

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In set theory, the characteristic function or indicator function of a subset X of a set S is the function, often denoted χA or IA, from S to the set {0,1} which takes the value 1 on elements of X and 0 otherwise.

We can express elementary set-theoretic operations in terms of characteristic functions:

  • Empty set:
  • Intersection:
  • Union:
  • complement:
  • Inclusion:

In mathematics, characteristic function can refer also to any several distinct concepts:

where "E" means expected value. See characteristic function (probability theory).