characteristic function

Definition Suppose $A$ is a subset of a set $X$ . Then the function

is the characteristic function for $A$ .

Properties

Suppose $A,B$ are subsets of a set $X$ .

1. For set intersections and set unions, we have \begin{eqnarray*} \chi_{A\cap B} &=& \chi_A \chi_B, \\ \chi_{A\cup B} &=& \chi_A + \chi_B - \chi_{A\cap B},\\ \chi_{A\cap B} &=& \min(\chi_A,\chi_B),\\ \chi_{A\cup B} &=& \max(\chi_A,\chi_B). \end{eqnarray*}
2. For the symmetric difference, $$\chi_{A\bigtriangleup B} = \chi_A + \chi_B - 2\chi_{A\cap B}.$$
3. For the set complement, $$\chi_{A^\complement} = 1-\chi_A. $$

Remarks

A synonym for characteristic function is indicator function [1].

Bibliography

1

G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.