characteristic function

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

\chi_{A}(x)=\begin{cases}1,&\text{when }x\in A,\\ 0,&\text{when }x\in X\setminus A\end{cases}

0.0.1 Properties

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

1.

For set intersections and set unions, we have

$\displaystyle\chi_{A\cap B}$ $\displaystyle=$ $\displaystyle\chi_{A}\chi_{B},$

$\displaystyle\chi_{A\cup B}$ $\displaystyle=$ $\displaystyle\chi_{A}+\chi_{B}-\chi_{A\cap B},$

$\displaystyle\chi_{A\cap B}$ $\displaystyle=$ $\displaystyle\min(\chi_{A},\chi_{B}),$

$\displaystyle\chi_{A\cup B}$ $\displaystyle=$ $\displaystyle\max(\chi_{A},\chi_{B}).$
2.

For the symmetric difference,

$\chi_{A\bigtriangleup B}=\chi_{A}+\chi_{B}-2\chi_{A\cap B}.$
3.

For the set complement,

$\chi_{A^{\complement}}=1-\chi_{A}.$

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

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

$\displaystyle\chi_{A\cap B}$	$\displaystyle=$	$\displaystyle\chi_{A}\chi_{B},$
$\displaystyle\chi_{A\cup B}$	$\displaystyle=$	$\displaystyle\chi_{A}+\chi_{B}-\chi_{A\cap B},$
$\displaystyle\chi_{A\cap B}$	$\displaystyle=$	$\displaystyle\min(\chi_{A},\chi_{B}),$
$\displaystyle\chi_{A\cup B}$	$\displaystyle=$	$\displaystyle\max(\chi_{A},\chi_{B}).$