Characteristic function

From Citizendium, the Citizens' Compendium

In set theory, the characteristic function or indicator function of a subset X of a set S is the function, often denoted χ_A or I_A, 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:

• The characteristic function in convex analysis:

• The characteristic state function in statistical mechanics.

• In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:

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

• The characteristic polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

• The characteristic function in game theory.