Characteristic function

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: $\chi _{\emptyset }=0;\,$
• Intersection: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{A \cap B} = \min\{\chi_A,\chi_B\} = \chi_A \cdot \chi_B ;\,}
• Union: $\chi _{A\cup B}=\max\{\chi _{A},\chi _{B}\}=\chi _{A}+\chi _{B}-\chi _{A}\cdot \chi _{B};\,$
• Symmetric difference: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{A \bigtriangleup B} = \chi_A + \chi_B \pmod 2 ;\,}
• Inclusion: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B \Leftrightarrow \chi_A \le \chi_B .\,}

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

• The characteristic function in convex analysis:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}}

• 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:

$\varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right)\,$

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.