# characteristic function

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

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

is the characteristic function for $A$.

## 0.0.1 Properties

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

1. 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. 2.

For the symmetric difference,

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

For the set complement,

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

## 0.0.2 Remarks

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

## References

• 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title characteristic function CharacteristicFunction 2013-03-22 11:48:31 2013-03-22 11:48:31 bbukh (348) bbukh (348) 12 bbukh (348) Definition msc 03-00 msc 26-00 msc 26A09 msc 28-00 indicator function SimpleFunction